Unique continuation for the Helmholtz equation using stabilized finite element methods

Mihai Nechita
(original), Finite element (FEM), html, ISI/JCR, Numerical Analysis, paper

Abstract

In this work we consider the computational approximation of a unique continuation problem for the Helmholtz equation using a stabilized finite element method. First conditional stability estimates are derived for which, under a convexity assumption on the geometry, the constants grow at most linearly in the wave number. Then these estimates are used to obtain error bounds for the finite element method that are explicit with respect to the wave number. Some numerical illustrations are given.

Authors

Erik Burman
Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, United Kingdom

Mihai Nechita
Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, United Kingdom

Lauri Oksanen
Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, United Kingdom

Keywords

Helmholtz equation; unique continuation; Finite Element method; wave number explicit; conditional Hölder stability.

Paper coordinates

E. Burman, M. Nechita, L. Oksanen, Unique continuation for the Helmholtz equation using stabilized finite element methods, J. Math. Pures Appl., 129 (2019), pp. 1-22.
DOI: 10.1016/j.matpur.2018.10.003

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Open Access: https://doi.org/10.1016/j.matpur.2018.10.003

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Journal de Mathématiques Pures et Appliquées

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Elsevier

DOI

10.1016/j.matpur.2018.10.003

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0021-7824

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google scholar citations

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UNIQUE CONTINUATION FOR THE HELMHOLTZ EQUATION USING STABILIZED FINITE ELEMENT METHODS

ERIK BURMAN, MIHAI NECHITA, AND LAURI OKSANEN

Abstract

In this work we consider the computational approximation of a unique continuation problem for the Helmholtz equation using a stabilized finite element method. First conditional stability estimates are derived for which, under a convexity assumption on the geometry, the constants grow at most linearly in the wave number. Then these estimates are used to obtain error bounds for the finite element method that are explicit with respect to the wave number. Some numerical illustrations are given.

1. Introduction

We consider a unique continuation (or data assimilation) problem for the Helmholtz equation
(1) Δ u + k 2 u = − f (1) Δ u + k 2 u = − f {:(1)Delta u+k^(2)u=-f:}\begin{equation*} \Delta u+k^{2} u=-f \tag{1} \end{equation*}(1)Δu+k2u=−f
and introduce a stabilized finite element method (FEM) to solve the problem computationally. Such methods have been previously studied for Poisson's equation in [5], [6] and [8], and for the heat equation in [10]. The main novelty of the present paper is that our method is robust with respect to the wave number k k kkk, and we prove convergence estimates with explicit dependence on k k kkk, see Theorem 1 and Theorem 2 below.
An abstract form of a unique continuation problem is as follows. Let ω ⊂ B ⊂ Ω ω ⊂ B ⊂ Ω omega sub B sub Omega\omega \subset B \subset \Omegaω⊂B⊂Ω be open, connected and non-empty sets in R 1 + n R 1 + n R^(1+n)\mathbb{R}^{1+n}R1+n and suppose that u ∈ H 2 ( Ω ) u ∈ H 2 ( Ω ) u inH^(2)(Omega)u \in H^{2}(\Omega)u∈H2(Ω) satisfies (1) in Ω Ω Omega\OmegaΩ. Given u u uuu in ω ω omega\omegaω and f f fff in Ω Ω Omega\OmegaΩ, find u u uuu in B B BBB.
This problem is non-trivial since no information on the boundary ∂ Ω ∂ Ω del Omega\partial \Omega∂Ω is given. It is well known, see e.g. [20], that if B ∖ ω ― ⊂ Ω B ∖ ω ¯ ⊂ Ω bar(B\\omega)sub Omega\overline{B \backslash \omega} \subset \OmegaB∖ω―⊂Ω then the problem is conditionally Hölder stable: for all k ≥ 0 k ≥ 0 k >= 0k \geq 0k≥0 there are C > 0 C > 0 C > 0C>0C>0 and α ∈ ( 0 , 1 ) α ∈ ( 0 , 1 ) alpha in(0,1)\alpha \in(0,1)α∈(0,1) such that for all u ∈ H 2 ( Ω ) u ∈ H 2 ( Ω ) u inH^(2)(Omega)u \in H^{2}(\Omega)u∈H2(Ω)
(2) ‖ u ‖ H 1 ( B ) ≤ C ( ‖ u ‖ H 1 ( ω ) + ‖ Δ u + k 2 u ‖ L 2 ( Ω ) ) α ‖ u ‖ H 1 ( Ω ) 1 − α (2) ‖ u ‖ H 1 ( B ) ≤ C ‖ u ‖ H 1 ( ω ) + Δ u + k 2 u L 2 ( Ω ) α ‖ u ‖ H 1 ( Ω ) 1 − α {:(2)||u||_(H^(1)(B)) <= C(||u||_(H^(1)(omega))+||Delta u+k^(2)u||_(L^(2)(Omega)))^(alpha)||u||_(H^(1)(Omega))^(1-alpha):}\begin{equation*} \|u\|_{H^{1}(B)} \leq C\left(\|u\|_{H^{1}(\omega)}+\left\|\Delta u+k^{2} u\right\|_{L^{2}(\Omega)}\right)^{\alpha}\|u\|_{H^{1}(\Omega)}^{1-\alpha} \tag{2} \end{equation*}(2)‖u‖H1(B)≤C(‖u‖H1(ω)+‖Δu+k2u‖L2(Ω))α‖u‖H1(Ω)1−α
If B ∖ ω B ∖ ω B\\omegaB \backslash \omegaB∖ω touches the boundary of Ω Ω Omega\OmegaΩ, then one can only expect logarithmic stability, since it was shown in the classical paper [21] that the optimal stability estimate for analytic continuation from a disk of radius strictly less than 1 to the concentric unit disk is of logarithmic type, and analytic functions are harmonic.
In general, the constants C C CCC and α α alpha\alphaα in (2) depend on k k kkk, as can be seen in Example 4 given in A. However, under suitable convexity assumptions on the geometry and direction of continuation it is possible to prove that in (2) both the constants C C CCC and α α alpha\alphaα are independent
of k k kkk, see the uniform estimate in Corollary 2 below, which is closely related to the so-called increased stability for unique continuation [17]. Obtaining optimal error bounds in the finite element approximation crucially depends on deriving estimates similar to (2), with weaker norms in the right-hand side, as in Corollary 3below, or in both sides, by shifting the Sobolev indices one degree down, as in Lemma 2 below.
In addition to robustness with respect to k k kkk, an advantage of using stabilized FEM for this unique continuation problem is that-when designed carefully-its implementation does not require information on the constants C C CCC and α α alpha\alphaα in (2), or any other quantity from the continuous stability theory, such as a specific choice of a Carleman weight function. Moreover, unlike other techniques such as Tikhonov regularization or quasi-reversibility, no auxiliary regularization parameters need to be introduced. The only asymptotic parameter in our method is the size of the finite element mesh, and in particular, we do not need to saturate the finite element method with respect to an auxiliary parameter as, for example, in the estimate (34) in 4.
Throughout the paper, C C CCC will denote a positive constant independent of the wave number k k kkk and the mesh size h h hhh, and which depends only on the geometry of the problem. By A ≲ B A ≲ B A≲BA \lesssim BA≲B we denote the inequality A ≤ C B A ≤ C B A <= CBA \leq C BA≤CB, where C C CCC is as above.
For the well-posed problem of the Helmholtz equation with the Robin boundary condition
(3) Δ u + k 2 u = − f in Ω and ∂ n u + i k u = 0 on ∂ Ω , (3) Δ u + k 2 u = − f  in  Ω  and  ∂ n u + i k u = 0  on  ∂ Ω , {:(3)Delta u+k^(2)u=-f quad" in "Omegaquad" and "quaddel_(n)u+iku=0quad" on "del Omega",":}\begin{equation*} \Delta u+k^{2} u=-f \quad \text { in } \Omega \quad \text { and } \quad \partial_{n} u+\mathrm{i} k u=0 \quad \text { on } \partial \Omega, \tag{3} \end{equation*}(3)Δu+k2u=−f in Ω and ∂nu+iku=0 on ∂Ω,
the following sharp bounds
(4) ‖ ∇ u ‖ L 2 ( Ω ) + k ‖ u ‖ L 2 ( Ω ) ≤ C ‖ f ‖ L 2 ( Ω ) (4) ‖ ∇ u ‖ L 2 ( Ω ) + k ‖ u ‖ L 2 ( Ω ) ≤ C ‖ f ‖ L 2 ( Ω ) {:(4)||grad u||_(L^(2)(Omega))+k||u||_(L^(2)(Omega)) <= C||f||_(L^(2)(Omega)):}\begin{equation*} \|\nabla u\|_{L^{2}(\Omega)}+k\|u\|_{L^{2}(\Omega)} \leq C\|f\|_{L^{2}(\Omega)} \tag{4} \end{equation*}(4)‖∇u‖L2(Ω)+k‖u‖L2(Ω)≤C‖f‖L2(Ω)
and
(5) ‖ u ‖ H 2 ( Ω ) ≤ C k ‖ f ‖ L 2 ( Ω ) (5) ‖ u ‖ H 2 ( Ω ) ≤ C k ‖ f ‖ L 2 ( Ω ) {:(5)||u||_(H^(2)(Omega)) <= Ck||f||_(L^(2)(Omega)):}\begin{equation*} \|u\|_{H^{2}(\Omega)} \leq C k\|f\|_{L^{2}(\Omega)} \tag{5} \end{equation*}(5)‖u‖H2(Ω)≤Ck‖f‖L2(Ω)
hold for a star-shaped Lipschitz domain Ω Ω Omega\OmegaΩ and any wave number k k kkk bounded away from zero [3. The error estimates that we derive in Section 3, e.g. ‖ u − u h ‖ H 1 ( B ) ≤ C ( h k ) α ‖ u ‖ ∗ u − u h H 1 ( B ) ≤ C ( h k ) α ‖ u ‖ ∗ ||u-u_(h)||_(H^(1)(B)) <= C(hk)^(alpha)||u||_(**)\left\|u-u_{h}\right\|_{H^{1}(B)} \leq C(h k)^{\alpha}\|u\|_{*}‖u−uh‖H1(B)≤C(hk)α‖u‖∗ in Theorem 2, contain the term
(6) ‖ u ‖ ∗ = ‖ u ‖ H 2 ( Ω ) + k 2 ‖ u ‖ L 2 ( Ω ) (6) ‖ u ‖ ∗ = ‖ u ‖ H 2 ( Ω ) + k 2 ‖ u ‖ L 2 ( Ω ) {:(6)||u||_(**)=||u||_(H^(2)(Omega))+k^(2)||u||_(L^(2)(Omega)):}\begin{equation*} \|u\|_{*}=\|u\|_{H^{2}(\Omega)}+k^{2}\|u\|_{L^{2}(\Omega)} \tag{6} \end{equation*}(6)‖u‖∗=‖u‖H2(Ω)+k2‖u‖L2(Ω)
which corresponds to the well-posed case term k ‖ f ‖ L 2 ( Ω ) k ‖ f ‖ L 2 ( Ω ) k||f||_(L^(2)(Omega))k\|f\|_{L^{2}(\Omega)}k‖f‖L2(Ω).
It is well known from the seminal works [2, 18, 19] that the finite element approximation of the Helmholtz problem is challenging also in the well-posed case due to the so-called pollution error. Indeed, to observe optimal convergence orders of H 1 H 1 H^(1)H^{1}H1 - and L 2 L 2 L^(2)L^{2}L2-errors the mesh size h h hhh must satisfy a smallness condition related to the wave number k k kkk, typically for piecewise affine elements, the condition k 2 h ≲ 1 k 2 h ≲ 1 k^(2)h≲1k^{2} h \lesssim 1k2h≲1. This is due to the dispersion error that is most important for low order approximation spaces. The situation improves if higher order polynomial approximation is used. Recently, the precise conditions for optimal convergence when using h p h p hph php-refinement ( p p ppp denotes the polynomial order of the approximation space) were shown in [24]. Under the assumption that the solution operator for Helmholtz problems is polynomially bounded in k k kkk, it is shown that quasi-optimality is obtained under the conditions that k h / p k h / p kh//pk h / pkh/p is sufficiently small and the polynomial degree p p ppp is at least O ( log ⁡ k ) O ( log ⁡ k ) O(log k)O(\log k)O(log⁡k).
Another way to obtain absolute stability (i.e. stability without, or under mild, conditions on the mesh size) of the approximate scheme is to use stabilization. The continuous interior
penalty stabilization (CIP) was introduced for the Helmholtz problem in [26], where stability was shown in the k h ≲ 1 k h ≲ 1 kh≲1k h \lesssim 1kh≲1 regime, and was subsequently used to obtain error bounds for standard piecewise affine elements when k 3 h 2 ≲ 1 k 3 h 2 ≲ 1 k^(3)h^(2)≲1k^{3} h^{2} \lesssim 1k3h2≲1. It was then shown in [11 that, in the one dimensional case, the CIP stabilization can also be used to eliminate the pollution error, provided the penalty parameter is appropriately chosen. When deriving error estimates for the stabilized FEM that we herein introduce, we shall make use of the mild condition k h ≲ 1 k h ≲ 1 kh≲1k h \lesssim 1kh≲1. To keep down the technical detail we restrict the analysis to the case of piecewise affine finite element spaces, but the extension of the proposed method to the high order case follows using the stabilization operators suggested in 5 (see also [7] for a discussion of the analysis in the ill-posed case).
From the point of view of applications, unique continuation problems often arise in control theory and inverse scattering problems. For instance, the above problem could arise when the acoustic wave field u u uuu is measured on ω ω omega\omegaω and there are unknown scatterers present outside Ω Ω Omega\OmegaΩ.

2. Continuum stability estimates

Our stabilized FEM will build on certain variations of the basic estimate (2), with the constants independent of the wave number, and we derive these estimates in the present section. The proofs are based on a Carleman estimate that is a variation of [17, Lemma 2.2] but we give a self-contained proof for the convenience of the reader. In 17] the Carleman estimate was used to derive a so-called increased stability estimate under suitable convexity assumptions on the geometry. To be more precise, let Γ ⊂ ∂ Ω Γ ⊂ ∂ Ω Gamma sub del Omega\Gamma \subset \partial \OmegaΓ⊂∂Ω be such that Γ ⊂ ∂ ω Γ ⊂ ∂ ω Gamma sub del omega\Gamma \subset \partial \omegaΓ⊂∂ω and Γ Γ Gamma\GammaΓ is at some positive distance away from ∂ ω ∩ Ω ∂ ω ∩ Ω del omega nn Omega\partial \omega \cap \Omega∂ω∩Ω. For a compact subset S S SSS of the open set Ω Ω Omega\OmegaΩ, let P ( ν ; d ) P ( ν ; d ) P(nu;d)P(\nu ; d)P(ν;d) denote the half space which has distance d d ddd from S S SSS and ν ν nu\nuν as the exterior normal vector. Let Ω ( ν ; d ) = P ( ν ; d ) ∩ Ω Ω ( ν ; d ) = P ( ν ; d ) ∩ Ω Omega(nu;d)=P(nu;d)nn Omega\Omega(\nu ; d)=P(\nu ; d) \cap \OmegaΩ(ν;d)=P(ν;d)∩Ω and denote by B B BBB the union of the sets Ω ( ν ; d ) Ω ( ν ; d ) Omega(nu;d)\Omega(\nu ; d)Ω(ν;d) over all ν ν nu\nuν for which P ( ν ; d ) ∩ ∂ Ω ⊂ Γ P ( ν ; d ) ∩ ∂ Ω ⊂ Γ P(nu;d)nn del Omega sub GammaP(\nu ; d) \cap \partial \Omega \subset \GammaP(ν;d)∩∂Ω⊂Γ. This geometric setting is exemplified by Figure 3a and it is illustrated in a general way in Figures 1 and 2 of [17] where B B BBB is denoted by Ω ( Γ ; d ) Ω ( Γ ; d ) Omega(Gamma;d)\Omega(\Gamma ; d)Ω(Γ;d). Under these assumptions it was proven that
(7) ‖ u ‖ L 2 ( B ) ≤ C F + C k − 1 F α ‖ u ‖ H 1 ( Ω ) 1 − α (7) ‖ u ‖ L 2 ( B ) ≤ C F + C k − 1 F α ‖ u ‖ H 1 ( Ω ) 1 − α {:(7)||u||_(L^(2)(B)) <= CF+Ck^(-1)F^(alpha)||u||_(H^(1)(Omega))^(1-alpha):}\begin{equation*} \|u\|_{L^{2}(B)} \leq C F+C k^{-1} F^{\alpha}\|u\|_{H^{1}(\Omega)}^{1-\alpha} \tag{7} \end{equation*}(7)‖u‖L2(B)≤CF+Ck−1Fα‖u‖H1(Ω)1−α
where F = ‖ u ‖ H 1 ( ω ) + ‖ Δ u + k 2 u ‖ L 2 ( Ω ) F = ‖ u ‖ H 1 ( ω ) + Δ u + k 2 u L 2 ( Ω ) F=||u||_(H^(1)(omega))+||Delta u+k^(2)u||_(L^(2)(Omega))F=\|u\|_{H^{1}(\omega)}+\left\|\Delta u+k^{2} u\right\|_{L^{2}(\Omega)}F=‖u‖H1(ω)+‖Δu+k2u‖L2(Ω) and the constants C C CCC and α α alpha\alphaα are independent of k k kkk. Here F F FFF can be interpreted as the size of the data in the unique continuation problem and the H 1 H 1 H^(1)H^{1}H1-norm of u u uuu as an a priori bound. As k k kkk grows, the first term on the right-hand side of (7) dominates the second one, and the stability is increasing in this sense.
As our focus is on designing a finite element method, we prefer to measure the size of the data in the weaker norm
E = ‖ u ‖ L 2 ( ω ) + ‖ Δ u + k 2 u ‖ H − 1 ( Ω ) . E = ‖ u ‖ L 2 ( ω ) + Δ u + k 2 u H − 1 ( Ω ) . E=||u||_(L^(2)(omega))+||Delta u+k^(2)u||_(H^(-1)(Omega)).E=\|u\|_{L^{2}(\omega)}+\left\|\Delta u+k^{2} u\right\|_{H^{-1}(\Omega)} .E=‖u‖L2(ω)+‖Δu+k2u‖H−1(Ω).
Taking u u uuu to be a plane wave solution to (1) suggests that
‖ u ‖ L 2 ( B ) ≤ C k E + C E α ‖ u ‖ L 2 ( Ω ) 1 − α ‖ u ‖ L 2 ( B ) ≤ C k E + C E α ‖ u ‖ L 2 ( Ω ) 1 − α ||u||_(L^(2)(B)) <= CkE+CE^(alpha)||u||_(L^(2)(Omega))^(1-alpha)\|u\|_{L^{2}(B)} \leq C k E+C E^{\alpha}\|u\|_{L^{2}(\Omega)}^{1-\alpha}‖u‖L2(B)≤CkE+CEα‖u‖L2(Ω)1−α
could be the right analogue of (7) when both the data and the a priori bound are in weaker norms. We show below, see Lemma 2, a stronger estimate with only the second term on the right-hand side.
Lemma 1 below captures the main step of the proof of our Carleman estimate. This is an elementary, but somewhat tedious, computation that establishes an identity similar to that in [23] where the constant in a Carleman estimate for the wave equation was studied. For an overview of Carleman estimates see [22, 25], the classical references are [15, Chapter 17] for second order elliptic equations, and [16, Chapter 28] for hyperbolic and more general equations. In the proofs, the idea is to use an exponential weight function e ℓ ( x ) e ℓ ( x ) e^(ℓ(x))e^{\ell(x)}eℓ(x) and study the expression
Δ ( e ℓ w ) = e ℓ Δ w + lower order terms Δ e ℓ w = e ℓ Δ w +  lower order terms  Delta(e^(ℓ)w)=e^(ℓ)Delta w+" lower order terms "\Delta\left(e^{\ell} w\right)=e^{\ell} \Delta w+\text { lower order terms }Δ(eℓw)=eℓΔw+ lower order terms 
or the conjugated operator e − ℓ Δ e ℓ e − ℓ Δ e ℓ e^(-ℓ)Deltae^(ℓ)e^{-\ell} \Delta e^{\ell}e−ℓΔeℓ. A typical approach is to study commutator estimates for the real and imaginary part of the principal symbol of the conjugated operator, see e.g. [22]. This can be seen as an alternative way to estimate the cross terms appearing in the proof of Lemma 1. Sometimes semiclassical analysis is used to derive the estimates, see e.g. [22]. This is very convenient when the estimates are shifted in the Sobolev scale, and we will use these techniques in Section 2.2 below.
2.1. A Carleman estimate and conditional Hölder stability. Denote by ( ⋅ , ⋅ ) , | ⋅ | ( ⋅ , ⋅ ) , | ⋅ | (*,*),|*|(\cdot, \cdot),|\cdot|(⋅,⋅),|⋅|, div , ∇ div , ∇ div,grad\operatorname{div}, \nabladiv,∇ and D 2 D 2 D^(2)D^{2}D2 the inner product, norm, divergence, gradient and Hessian with respect to the Euclidean structure in Ω ⊂ R 1 + n Ω ⊂ R 1 + n Omega subR^(1+n)\Omega \subset \mathbb{R}^{1+n}Ω⊂R1+n. (Below, Lemma 1 and Corollary 1 are written so that they hold also when Ω Ω Omega\OmegaΩ is a Riemannian manifold and the above concepts are replaced with their Riemannian analogues.)
Lemma 1. Let k ≥ 0 k ≥ 0 k >= 0k \geq 0k≥0. Let ℓ , w ∈ C 2 ( Ω ) ℓ , w ∈ C 2 ( Ω ) ℓ,w inC^(2)(Omega)\ell, w \in C^{2}(\Omega)ℓ,w∈C2(Ω) and σ ∈ C 1 ( Ω ) σ ∈ C 1 ( Ω ) sigma inC^(1)(Omega)\sigma \in C^{1}(\Omega)σ∈C1(Ω). We define v = e ℓ w v = e ℓ w v=e^(ℓ)wv=e^{\ell} wv=eℓw, and
a = σ − Δ ℓ , q = k 2 + a + | ∇ ℓ | 2 , b = − σ v − 2 ( ∇ v , ∇ ℓ ) , c = ( | ∇ v | 2 − q v 2 ) ∇ ℓ . a = σ − Δ ℓ , q = k 2 + a + | ∇ ℓ | 2 , b = − σ v − 2 ( ∇ v , ∇ ℓ ) , c = | ∇ v | 2 − q v 2 ∇ ℓ . a=sigma-Deltaℓ,quad q=k^(2)+a+|gradℓ|^(2),quad b=-sigma v-2(grad v,gradℓ),quad c=(|grad v|^(2)-qv^(2))gradℓ.a=\sigma-\Delta \ell, \quad q=k^{2}+a+|\nabla \ell|^{2}, \quad b=-\sigma v-2(\nabla v, \nabla \ell), \quad c=\left(|\nabla v|^{2}-q v^{2}\right) \nabla \ell .a=σ−Δℓ,q=k2+a+|∇ℓ|2,b=−σv−2(∇v,∇ℓ),c=(|∇v|2−qv2)∇ℓ.
Then
e 2 ℓ ( Δ w + k 2 w ) 2 / 2 = ( Δ v + q v ) 2 / 2 + b 2 / 2 + a | ∇ v | 2 + 2 D 2 ℓ ( ∇ v , ∇ v ) + ( − a | ∇ ℓ | 2 + 2 D 2 ℓ ( ∇ ℓ , ∇ ℓ ) ) v 2 − k 2 a v 2 + div ( b ∇ v + c ) + R e 2 ℓ Δ w + k 2 w 2 / 2 = ( Δ v + q v ) 2 / 2 + b 2 / 2 + a | ∇ v | 2 + 2 D 2 ℓ ( ∇ v , ∇ v ) + − a | ∇ ℓ | 2 + 2 D 2 ℓ ( ∇ ℓ , ∇ ℓ ) v 2 − k 2 a v 2 + div ( b ∇ v + c ) + R {:[e^(2ℓ)(Delta w+k^(2)w)^(2)//2=(Delta v+qv)^(2)//2+b^(2)//2],[quad+a|grad v|^(2)+2D^(2)ℓ(grad v","grad v)+(-a|gradℓ|^(2)+2D^(2)ℓ(gradℓ,gradℓ))v^(2)-k^(2)av^(2)],[quad+div(b grad v+c)+R]:}\begin{aligned} & e^{2 \ell}\left(\Delta w+k^{2} w\right)^{2} / 2=(\Delta v+q v)^{2} / 2+b^{2} / 2 \\ & \quad+a|\nabla v|^{2}+2 D^{2} \ell(\nabla v, \nabla v)+\left(-a|\nabla \ell|^{2}+2 D^{2} \ell(\nabla \ell, \nabla \ell)\right) v^{2}-k^{2} a v^{2} \\ & \quad+\operatorname{div}(b \nabla v+c)+R \end{aligned}e2ℓ(Δw+k2w)2/2=(Δv+qv)2/2+b2/2+a|∇v|2+2D2ℓ(∇v,∇v)+(−a|∇ℓ|2+2D2ℓ(∇ℓ,∇ℓ))v2−k2av2+div(b∇v+c)+R
where R = ( ∇ σ , ∇ v ) v + ( div ( a ∇ ℓ ) − a σ ) v 2 R = ( ∇ σ , ∇ v ) v + ( div ( a ∇ ℓ ) − a σ ) v 2 R=(grad sigma,grad v)v+(div(a gradℓ)-a sigma)v^(2)R=(\nabla \sigma, \nabla v) v+(\operatorname{div}(a \nabla \ell)-a \sigma) v^{2}R=(∇σ,∇v)v+(div(a∇ℓ)−aσ)v2.
A proof of this result is given in A. In the present paper we use Lemma 1 only with the choice σ = Δ ℓ σ = Δ ℓ sigma=Deltaℓ\sigma=\Delta \ellσ=Δℓ, or equivalently a = 0 a = 0 a=0a=0a=0, but the more general version of the lemma is useful when non-convex geometries are considered. In fact, instead of using a strictly convex function ϕ ϕ phi\phiϕ as in Corollary 1 below, it is possible to use a function ϕ ϕ phi\phiϕ without critical points, and convexify by taking ℓ = τ e α ϕ ℓ = τ e α ϕ ℓ=taue^(alpha phi)\ell=\tau e^{\alpha \phi}ℓ=τeαϕ and σ = Δ ℓ + α λ ℓ σ = Δ ℓ + α λ ℓ sigma=Deltaℓ+alpha lambdaℓ\sigma=\Delta \ell+\alpha \lambda \ellσ=Δℓ+αλℓ for suitable constants α α alpha\alphaα and λ λ lambda\lambdaλ. In the present context this will lead to an estimate that is not robust with respect to k k kkk, but we will use such a technique in the forthcoming paper [9].
Corollary 1 (Pointwise Carleman estimate). Let ϕ ∈ C 3 ( Ω ) ϕ ∈ C 3 ( Ω ) phi inC^(3)(Omega)\phi \in C^{3}(\Omega)ϕ∈C3(Ω) be a strictly convex function without critical points, and choose ρ > 0 ρ > 0 rho > 0\rho>0ρ>0 such that
D 2 ϕ ( X , X ) ≥ ρ | X | 2 , X ∈ T x Ω , x ∈ Ω D 2 ϕ ( X , X ) ≥ ρ | X | 2 , X ∈ T x Ω , x ∈ Ω D^(2)phi(X,X) >= rho|X|^(2),quad X inT_(x)Omega,x in OmegaD^{2} \phi(X, X) \geq \rho|X|^{2}, \quad X \in T_{x} \Omega, x \in \OmegaD2ϕ(X,X)≥ρ|X|2,X∈TxΩ,x∈Ω
Let τ > 0 τ > 0 tau > 0\tau>0τ>0 and w ∈ C 2 ( Ω ) w ∈ C 2 ( Ω ) w inC^(2)(Omega)w \in C^{2}(\Omega)w∈C2(Ω). We define ℓ = τ ϕ , v = e ℓ w ℓ = τ ϕ , v = e ℓ w ℓ=tau phi,v=e^(ℓ)w\ell=\tau \phi, v=e^{\ell} wℓ=τϕ,v=eℓw, and
b = − ( Δ ℓ ) v − 2 ( ∇ v , ∇ ℓ ) , c = ( | ∇ v | 2 − ( k 2 + | ∇ ℓ | 2 ) v 2 ) ∇ ℓ b = − ( Δ ℓ ) v − 2 ( ∇ v , ∇ ℓ ) , c = | ∇ v | 2 − k 2 + | ∇ ℓ | 2 v 2 ∇ ℓ b=-(Deltaℓ)v-2(grad v,gradℓ),quad c=(|grad v|^(2)-(k^(2)+|gradℓ|^(2))v^(2))gradℓb=-(\Delta \ell) v-2(\nabla v, \nabla \ell), \quad c=\left(|\nabla v|^{2}-\left(k^{2}+|\nabla \ell|^{2}\right) v^{2}\right) \nabla \ellb=−(Δℓ)v−2(∇v,∇ℓ),c=(|∇v|2−(k2+|∇ℓ|2)v2)∇ℓ
Then
e 2 τ ϕ ( ( a 0 τ − b 0 ) τ 2 w 2 + ( a 1 τ − b 1 ) | ∇ w | 2 ) + div ( b ∇ v + c ) ≤ e 2 τ ϕ ( Δ w + k 2 w ) 2 / 2 , e 2 τ ϕ a 0 τ − b 0 τ 2 w 2 + a 1 τ − b 1 | ∇ w | 2 + div ( b ∇ v + c ) ≤ e 2 τ ϕ Δ w + k 2 w 2 / 2 , e^(2tau phi)((a_(0)tau-b_(0))tau^(2)w^(2)+(a_(1)tau-b_(1))|grad w|^(2))+div(b grad v+c) <= e^(2tau phi)(Delta w+k^(2)w)^(2)//2,e^{2 \tau \phi}\left(\left(a_{0} \tau-b_{0}\right) \tau^{2} w^{2}+\left(a_{1} \tau-b_{1}\right)|\nabla w|^{2}\right)+\operatorname{div}(b \nabla v+c) \leq e^{2 \tau \phi}\left(\Delta w+k^{2} w\right)^{2} / 2,e2τϕ((a0τ−b0)τ2w2+(a1τ−b1)|∇w|2)+div(b∇v+c)≤e2τϕ(Δw+k2w)2/2,
where the constants a j , b j > 0 , j = 0 , 1 a j , b j > 0 , j = 0 , 1 a_(j),b_(j) > 0,j=0,1a_{j}, b_{j}>0, j=0,1aj,bj>0,j=0,1, depend only on ρ , inf x ∈ Ω | ∇ ϕ ( x ) | 2 ρ , inf x ∈ Ω   | ∇ ϕ ( x ) | 2 rho,i n f_(x in Omega)|grad phi(x)|^(2)\rho, \inf _{x \in \Omega}|\nabla \phi(x)|^{2}ρ,infx∈Ω|∇ϕ(x)|2 and sup x ∈ Ω | ∇ ( Δ ϕ ( x ) ) | 2 sup x ∈ Ω   | ∇ ( Δ ϕ ( x ) ) | 2 s u p_(x in Omega)|grad(Delta phi(x))|^(2)\sup _{x \in \Omega}|\nabla(\Delta \phi(x))|^{2}supx∈Ω|∇(Δϕ(x))|2.
Proof. We employ the equality in Lemma 1 with ℓ = τ ϕ ℓ = τ ϕ ℓ=tau phi\ell=\tau \phiℓ=τϕ and σ = Δ ℓ σ = Δ ℓ sigma=Deltaℓ\sigma=\Delta \ellσ=Δℓ. With this choice of σ σ sigma\sigmaσ, it holds that a = 0 a = 0 a=0a=0a=0. As the two first terms on the right-hand side of the equality are positive, it is enough to consider
2 D 2 ℓ ( ∇ v , ∇ v ) + 2 D 2 ℓ ( ∇ ℓ , ∇ ℓ ) v 2 + R ≥ 2 ρ τ | ∇ v | 2 + 2 ρ τ 3 | ∇ ϕ | 2 v 2 − τ | ∇ ( Δ ϕ ) | | ∇ v | | v | . 2 D 2 ℓ ( ∇ v , ∇ v ) + 2 D 2 ℓ ( ∇ ℓ , ∇ ℓ ) v 2 + R ≥ 2 ρ τ | ∇ v | 2 + 2 ρ τ 3 | ∇ ϕ | 2 v 2 − τ | ∇ ( Δ ϕ ) | | ∇ v | | v | . {:[2D^(2)ℓ(grad v","grad v)+2D^(2)ℓ(gradℓ","gradℓ)v^(2)+R],[quad >= 2rho tau|grad v|^(2)+2rhotau^(3)|grad phi|^(2)v^(2)-tau|grad(Delta phi)||grad v||v|.]:}\begin{aligned} & 2 D^{2} \ell(\nabla v, \nabla v)+2 D^{2} \ell(\nabla \ell, \nabla \ell) v^{2}+R \\ & \quad \geq 2 \rho \tau|\nabla v|^{2}+2 \rho \tau^{3}|\nabla \phi|^{2} v^{2}-\tau|\nabla(\Delta \phi)||\nabla v||v| . \end{aligned}2D2ℓ(∇v,∇v)+2D2ℓ(∇ℓ,∇ℓ)v2+R≥2ρτ|∇v|2+2ρτ3|∇ϕ|2v2−τ|∇(Δϕ)||∇v||v|.
The claim follows by combining this with
| ∇ v | 2 = e 2 τ ϕ | τ w ∇ ϕ + ∇ w | 2 ≥ e 2 τ ϕ 1 3 | ∇ w | 2 − e 2 τ ϕ 1 2 | ∇ ϕ | 2 τ 2 w 2 , | ∇ v | 2 = e 2 τ ϕ | τ w ∇ ϕ + ∇ w | 2 ≥ e 2 τ ϕ 1 3 | ∇ w | 2 − e 2 τ ϕ 1 2 | ∇ ϕ | 2 τ 2 w 2 , |grad v|^(2)=e^(2tau phi)|tau w grad phi+grad w|^(2) >= e^(2tau phi)(1)/(3)|grad w|^(2)-e^(2tau phi)(1)/(2)|grad phi|^(2)tau^(2)w^(2),|\nabla v|^{2}=e^{2 \tau \phi}|\tau w \nabla \phi+\nabla w|^{2} \geq e^{2 \tau \phi} \frac{1}{3}|\nabla w|^{2}-e^{2 \tau \phi} \frac{1}{2}|\nabla \phi|^{2} \tau^{2} w^{2},|∇v|2=e2τϕ|τw∇ϕ+∇w|2≥e2τϕ13|∇w|2−e2τϕ12|∇ϕ|2τ2w2,
and
τ | ∇ ( Δ ϕ ) | | ∇ v | | v | ≤ C ( | ∇ v | 2 + τ 2 | v | 2 ) . τ | ∇ ( Δ ϕ ) | | ∇ v | | v | ≤ C | ∇ v | 2 + τ 2 | v | 2 . tau|grad(Delta phi)||grad v||v| <= C(|grad v|^(2)+tau^(2)|v|^(2)).\tau|\nabla(\Delta \phi)||\nabla v||v| \leq C\left(|\nabla v|^{2}+\tau^{2}|v|^{2}\right) .τ|∇(Δϕ)||∇v||v|≤C(|∇v|2+τ2|v|2). ◻ ◻ ◻\square◻
The above Carleman estimate implies an inequality that is similar to the three-ball inequality, see e.g. [1]. The main difference is that here the foliation along spheres is followed in the opposite direction, i.e. the convex direction.
When continuing the solution inside the convex hull of ω ω omega\omegaω as in [17], we consider for simplicity a specific geometric setting defined in Corollary 2 below and illustrated in Figure 1. The stability estimates we prove below in Corollary 2 and Corollary 3, and Lemma 2 also hold in other geometric settings in which B B BBB is included in the convex hull of ω ω omega\omegaω and B ∖ ω B ∖ ω B\\omegaB \backslash \omegaB∖ω does not touch the boundary of Ω Ω Omega\OmegaΩ, such as the one in Figure 3a. We prove this in Example 1.
Figure 1. The geometric setting in Corollary 2.
We use the following notation for a half space
H = { ( x 0 , … , x n ) ∈ R 1 + n ; x 0 < 0 } H = x 0 , … , x n ∈ R 1 + n ; x 0 < 0 H={(x^(0),dots,x^(n))inR^(1+n);x^(0) < 0}H=\left\{\left(x^{0}, \ldots, x^{n}\right) \in \mathbb{R}^{1+n} ; x^{0}<0\right\}H={(x0,…,xn)∈R1+n;x0<0}
Corollary 2. Let r > 0 , β > 0 , R > r r > 0 , β > 0 , R > r r > 0,beta > 0,R > rr>0, \beta>0, R>rr>0,β>0,R>r and r 2 + β 2 < ρ < R 2 + β 2 r 2 + β 2 < ρ < R 2 + β 2 sqrt(r^(2)+beta^(2)) < rho < sqrt(R^(2)+beta^(2))\sqrt{r^{2}+\beta^{2}}<\rho<\sqrt{R^{2}+\beta^{2}}r2+β2<ρ<R2+β2. Define y = ( β , 0 , … , 0 ) y = ( β , 0 , … , 0 ) y=(beta,0,dots,0)y= (\beta, 0, \ldots, 0)y=(β,0,…,0) and
Ω = H ∩ B ( 0 , R ) , ω = Ω ∖ B ( 0 , r ) ― , B = Ω ∖ B ( y , ρ ) ― Ω = H ∩ B ( 0 , R ) , ω = Ω ∖ B ( 0 , r ) ¯ , B = Ω ∖ B ( y , ρ ) ¯ Omega=H nn B(0,R),quad omega=Omega\\ bar(B(0,r)),quad B=Omega\\ bar(B(y,rho))\Omega=H \cap B(0, R), \quad \omega=\Omega \backslash \overline{B(0, r)}, \quad B=\Omega \backslash \overline{B(y, \rho)}Ω=H∩B(0,R),ω=Ω∖B(0,r)―,B=Ω∖B(y,ρ)―
Then there are C > 0 C > 0 C > 0C>0C>0 and α ∈ ( 0 , 1 ) α ∈ ( 0 , 1 ) alpha in(0,1)\alpha \in(0,1)α∈(0,1) such that for all u ∈ C 2 ( Ω ) u ∈ C 2 ( Ω ) u inC^(2)(Omega)u \in C^{2}(\Omega)u∈C2(Ω) and k ≥ 0 k ≥ 0 k >= 0k \geq 0k≥0
‖ u ‖ H 1 ( B ) ≤ C ( ‖ u ‖ H 1 ( ω ) + ‖ Δ u + k 2 u ‖ L 2 ( Ω ) ) α ‖ u ‖ H 1 ( Ω ) 1 − α ‖ u ‖ H 1 ( B ) ≤ C ‖ u ‖ H 1 ( ω ) + Δ u + k 2 u L 2 ( Ω ) α ‖ u ‖ H 1 ( Ω ) 1 − α ||u||_(H^(1)(B)) <= C(||u||_(H^(1)(omega))+||Delta u+k^(2)u||_(L^(2)(Omega)))^(alpha)||u||_(H^(1)(Omega))^(1-alpha)\|u\|_{H^{1}(B)} \leq C\left(\|u\|_{H^{1}(\omega)}+\left\|\Delta u+k^{2} u\right\|_{L^{2}(\Omega)}\right)^{\alpha}\|u\|_{H^{1}(\Omega)}^{1-\alpha}‖u‖H1(B)≤C(‖u‖H1(ω)+‖Δu+k2u‖L2(Ω))α‖u‖H1(Ω)1−α
Proof. Choose r 2 + β 2 < s < ρ r 2 + β 2 < s < ρ sqrt(r^(2)+beta^(2)) < s < rho\sqrt{r^{2}+\beta^{2}}<s<\rhor2+β2<s<ρ and observe that ∂ Ω ∖ B ( y , s ) ⊂ ω ¯ ∂ Ω ∖ B ( y , s ) ⊂ ω ¯ del Omega\\B(y,s)sub bar(omega)\partial \Omega \backslash B(y, s) \subset \bar{\omega}∂Ω∖B(y,s)⊂ω¯. Define ϕ ( x ) = | x − y | 2 ϕ ( x ) = | x − y | 2 phi(x)=|x-y|^(2)\phi(x)=|x-y|^{2}ϕ(x)=|x−y|2. Then ϕ ϕ phi\phiϕ is smooth and strictly convex in Ω ¯ Ω ¯ bar(Omega)\bar{\Omega}Ω¯, and it does not have critical points there.
Choose χ ∈ C 0 ∞ ( Ω ) χ ∈ C 0 ∞ ( Ω ) chi inC_(0)^(oo)(Omega)\chi \in C_{0}^{\infty}(\Omega)χ∈C0∞(Ω) such that χ = 1 χ = 1 chi=1\chi=1χ=1 in Ω ∖ ( B ( y , s ) ∪ ω ) Ω ∖ ( B ( y , s ) ∪ ω ) Omega\\(B(y,s)uu omega)\Omega \backslash(B(y, s) \cup \omega)Ω∖(B(y,s)∪ω) and set w = χ u w = χ u w=chi uw=\chi uw=χu. Corollary 1 implies that for large τ > 0 τ > 0 tau > 0\tau>0τ>0
(8) ∫ Ω ( τ 3 w 2 + τ | ∇ w | 2 ) e 2 τ ϕ d x ≤ C ∫ Ω ( Δ w + k 2 w ) 2 e 2 τ ϕ d x (8) ∫ Ω   τ 3 w 2 + τ | ∇ w | 2 e 2 τ ϕ d x ≤ C ∫ Ω   Δ w + k 2 w 2 e 2 τ ϕ d x {:(8)int_(Omega)(tau^(3)w^(2)+tau|grad w|^(2))e^(2tau phi)dx <= Cint_(Omega)(Delta w+k^(2)w)^(2)e^(2tau phi)dx:}\begin{equation*} \int_{\Omega}\left(\tau^{3} w^{2}+\tau|\nabla w|^{2}\right) e^{2 \tau \phi} d x \leq C \int_{\Omega}\left(\Delta w+k^{2} w\right)^{2} e^{2 \tau \phi} d x \tag{8} \end{equation*}(8)∫Ω(τ3w2+τ|∇w|2)e2τϕdx≤C∫Ω(Δw+k2w)2e2τϕdx
a result also stated, without a detailed proof, in [ 20 [ 20 [20[20[20, Exercise 3.4.6 ] ] ]]]. The commutator [ Δ , χ ] [ Δ , χ ] [Delta,chi][\Delta, \chi][Δ,χ] vanishes outside B ( y , s ) ∪ ω B ( y , s ) ∪ ω B(y,s)uu omegaB(y, s) \cup \omegaB(y,s)∪ω and ϕ < s 2 ϕ < s 2 phi < s^(2)\phi<s^{2}ϕ<s2 in B ( y , s ) B ( y , s ) B(y,s)B(y, s)B(y,s). Hence the right-hand side of (8) is bounded by a constant times
(9) ∫ Ω | Δ u + k 2 u | 2 e 2 τ ϕ d x + ∫ B ( y , s ) ∪ ω | [ Δ , χ ] u | 2 e 2 τ ϕ d x ≤ C e 2 τ ( β + R ) 2 ( ‖ Δ u + k 2 u ‖ L 2 ( Ω ) 2 + ‖ u ‖ H 1 ( ω ) 2 ) + C e 2 τ s 2 ‖ u ‖ H 1 ( B ( y , s ) ) 2 (9) ∫ Ω   Δ u + k 2 u 2 e 2 τ ϕ d x + ∫ B ( y , s ) ∪ ω   | [ Δ , χ ] u | 2 e 2 τ ϕ d x ≤ C e 2 τ ( β + R ) 2 Δ u + k 2 u L 2 ( Ω ) 2 + ‖ u ‖ H 1 ( ω ) 2 + C e 2 τ s 2 ‖ u ‖ H 1 ( B ( y , s ) ) 2 {:[(9)int_(Omega)|Delta u+k^(2)u|^(2)e^(2tau phi)dx+int_(B(y,s)uu omega)|[Delta","chi]u|^(2)e^(2tau phi)dx],[quad <= Ce^(2tau(beta+R)^(2))(||Delta u+k^(2)u||_(L^(2)(Omega))^(2)+||u||_(H^(1)(omega))^(2))+Ce^(2taus^(2))||u||_(H^(1)(B(y,s)))^(2)]:}\begin{align*} & \int_{\Omega}\left|\Delta u+k^{2} u\right|^{2} e^{2 \tau \phi} d x+\int_{B(y, s) \cup \omega}|[\Delta, \chi] u|^{2} e^{2 \tau \phi} d x \tag{9}\\ & \quad \leq C e^{2 \tau(\beta+R)^{2}}\left(\left\|\Delta u+k^{2} u\right\|_{L^{2}(\Omega)}^{2}+\|u\|_{H^{1}(\omega)}^{2}\right)+C e^{2 \tau s^{2}}\|u\|_{H^{1}(B(y, s))}^{2} \end{align*}(9)∫Ω|Δu+k2u|2e2τϕdx+∫B(y,s)∪ω|[Δ,χ]u|2e2τϕdx≤Ce2τ(β+R)2(‖Δu+k2u‖L2(Ω)2+‖u‖H1(ω)2)+Ce2τs2‖u‖H1(B(y,s))2
The left-hand side of (8) is bounded from below by
(10) ∫ B ( τ | ∇ u | 2 + τ 3 | u | 2 ) e 2 τ ϕ d x ≥ e 2 τ ρ 2 ‖ u ‖ H 1 ( B ) 2 (10) ∫ B   τ | ∇ u | 2 + τ 3 | u | 2 e 2 τ ϕ d x ≥ e 2 τ ρ 2 ‖ u ‖ H 1 ( B ) 2 {:(10)int_(B)(tau|grad u|^(2)+tau^(3)|u|^(2))e^(2tau phi)dx >= e^(2taurho^(2))||u||_(H^(1)(B))^(2):}\begin{equation*} \int_{B}\left(\tau|\nabla u|^{2}+\tau^{3}|u|^{2}\right) e^{2 \tau \phi} d x \geq e^{2 \tau \rho^{2}}\|u\|_{H^{1}(B)}^{2} \tag{10} \end{equation*}(10)∫B(τ|∇u|2+τ3|u|2)e2τϕdx≥e2τρ2‖u‖H1(B)2
The inequalities (8)-(10) imply
‖ u ‖ H 1 ( B ) ≤ e q τ ( ‖ Δ u + k 2 u ‖ L 2 ( Ω ) + ‖ u ‖ H 1 ( ω ) ) + e − p τ ‖ u ‖ H 1 ( Ω ) , ‖ u ‖ H 1 ( B ) ≤ e q τ Δ u + k 2 u L 2 ( Ω ) + ‖ u ‖ H 1 ( ω ) + e − p τ ‖ u ‖ H 1 ( Ω ) , ||u||_(H^(1)(B)) <= e^(q tau)(||Delta u+k^(2)u||_(L^(2)(Omega))+||u||_(H^(1)(omega)))+e^(-p tau)||u||_(H^(1)(Omega)),\|u\|_{H^{1}(B)} \leq e^{q \tau}\left(\left\|\Delta u+k^{2} u\right\|_{L^{2}(\Omega)}+\|u\|_{H^{1}(\omega)}\right)+e^{-p \tau}\|u\|_{H^{1}(\Omega)},‖u‖H1(B)≤eqτ(‖Δu+k2u‖L2(Ω)+‖u‖H1(ω))+e−pτ‖u‖H1(Ω),
where q = ( β + R ) 2 − ρ 2 q = ( β + R ) 2 − ρ 2 q=(beta+R)^(2)-rho^(2)q=(\beta+R)^{2}-\rho^{2}q=(β+R)2−ρ2 and p = ρ 2 − s 2 > 0 p = ρ 2 − s 2 > 0 p=rho^(2)-s^(2) > 0p=\rho^{2}-s^{2}>0p=ρ2−s2>0. The claim follows from [22, Lemma 5.2].
Corollary 3. Let ω ⊂ B ⊂ Ω ω ⊂ B ⊂ Ω omega sub B sub Omega\omega \subset B \subset \Omegaω⊂B⊂Ω be defined as in Corollary 2. Then there are C > 0 C > 0 C > 0C>0C>0 and α ∈ ( 0 , 1 ) α ∈ ( 0 , 1 ) alpha in(0,1)\alpha \in(0,1)α∈(0,1) such that
‖ u ‖ H 1 ( B ) ≤ C k ( ‖ u ‖ L 2 ( ω ) + ‖ Δ u + k 2 u ‖ H − 1 ( Ω ) ) α ( ‖ u ‖ L 2 ( Ω ) + ‖ Δ u + k 2 u ‖ H − 1 ( Ω ) ) 1 − α . ‖ u ‖ H 1 ( B ) ≤ C k ‖ u ‖ L 2 ( ω ) + Δ u + k 2 u H − 1 ( Ω ) α ‖ u ‖ L 2 ( Ω ) + Δ u + k 2 u H − 1 ( Ω ) 1 − α . ||u||_(H^(1)(B)) <= Ck(||u||_(L^(2)(omega))+||Delta u+k^(2)u||_(H^(-1)(Omega)))^(alpha)(||u||_(L^(2)(Omega))+||Delta u+k^(2)u||_(H^(-1)(Omega)))^(1-alpha).\|u\|_{H^{1}(B)} \leq C k\left(\|u\|_{L^{2}(\omega)}+\left\|\Delta u+k^{2} u\right\|_{H^{-1}(\Omega)}\right)^{\alpha}\left(\|u\|_{L^{2}(\Omega)}+\left\|\Delta u+k^{2} u\right\|_{H^{-1}(\Omega)}\right)^{1-\alpha} .‖u‖H1(B)≤Ck(‖u‖L2(ω)+‖Δu+k2u‖H−1(Ω))α(‖u‖L2(Ω)+‖Δu+k2u‖H−1(Ω))1−α.
Proof. Let ω 1 ⊂ ω ⊂ B ⊂ Ω 1 ⊂ Ω ω 1 ⊂ ω ⊂ B ⊂ Ω 1 ⊂ Ω omega_(1)sub omega sub B subOmega_(1)sub Omega\omega_{1} \subset \omega \subset B \subset \Omega_{1} \subset \Omegaω1⊂ω⊂B⊂Ω1⊂Ω, denote for brevity by L L L\mathcal{L}L the operator Δ + k 2 Δ + k 2 Delta+k^(2)\Delta+k^{2}Δ+k2, and consider the following auxiliary problem
L w = L u in Ω 1 ∂ n w + i k w = 0 on ∂ Ω 1 L w = L u  in  Ω 1 ∂ n w + i k w = 0  on  ∂ Ω 1 {:[Lw=Lu quad" in "Omega_(1)],[del_(n)w+ikw=0quad" on "delOmega_(1)]:}\begin{aligned} \mathcal{L} w & =\mathcal{L} u \quad \text { in } \Omega_{1} \\ \partial_{n} w+\mathrm{i} k w & =0 \quad \text { on } \partial \Omega_{1} \end{aligned}Lw=Lu in Ω1∂nw+ikw=0 on ∂Ω1
whose solution satisfies the estimate [3, Corollary 1.10]
‖ ∇ w ‖ L 2 ( Ω 1 ) + k ‖ w ‖ L 2 ( Ω 1 ) ≤ C k ‖ L u ‖ H − 1 ( Ω 1 ) ‖ ∇ w ‖ L 2 Ω 1 + k ‖ w ‖ L 2 Ω 1 ≤ C k ‖ L u ‖ H − 1 Ω 1 ||grad w||_(L^(2)(Omega_(1)))+k||w||_(L^(2)(Omega_(1))) <= Ck||Lu||_(H^(-1)(Omega_(1)))\|\nabla w\|_{L^{2}\left(\Omega_{1}\right)}+k\|w\|_{L^{2}\left(\Omega_{1}\right)} \leq C k\|\mathcal{L} u\|_{H^{-1}\left(\Omega_{1}\right)}‖∇w‖L2(Ω1)+k‖w‖L2(Ω1)≤Ck‖Lu‖H−1(Ω1)
which gives
‖ w ‖ H 1 ( Ω 1 ) ≤ C k ‖ L u ‖ H − 1 ( Ω ) ‖ w ‖ H 1 Ω 1 ≤ C k ‖ L u ‖ H − 1 ( Ω ) ||w||_(H^(1)(Omega_(1))) <= Ck||Lu||_(H^(-1)(Omega))\|w\|_{H^{1}\left(\Omega_{1}\right)} \leq C k\|\mathcal{L} u\|_{H^{-1}(\Omega)}‖w‖H1(Ω1)≤Ck‖Lu‖H−1(Ω)
For v = u − w v = u − w v=u-wv=u-wv=u−w we have L v = 0 L v = 0 Lv=0\mathcal{L} v=0Lv=0 in Ω 1 Ω 1 Omega_(1)\Omega_{1}Ω1. The stability estimate in Corollary 2 used for ω 1 , B , Ω 1 ω 1 , B , Ω 1 omega_(1),B,Omega_(1)\omega_{1}, B, \Omega_{1}ω1,B,Ω1 reads as
‖ v ‖ H 1 ( B ) ≤ C ‖ v ‖ H 1 ( ω 1 ) α ‖ v ‖ H 1 ( Ω 1 ) 1 − α ‖ v ‖ H 1 ( B ) ≤ C ‖ v ‖ H 1 ω 1 α ‖ v ‖ H 1 Ω 1 1 − α ||v||_(H^(1)(B)) <= C||v||_(H^(1)(omega_(1)))^(alpha)||v||_(H^(1)(Omega_(1)))^(1-alpha)\|v\|_{H^{1}(B)} \leq C\|v\|_{H^{1}\left(\omega_{1}\right)}^{\alpha}\|v\|_{H^{1}\left(\Omega_{1}\right)}^{1-\alpha}‖v‖H1(B)≤C‖v‖H1(ω1)α‖v‖H1(Ω1)1−α
and the following estimates hold
‖ u ‖ H 1 ( B ) ≤ ‖ v ‖ H 1 ( B ) + ‖ w ‖ H 1 ( B ) ≤ C ( ‖ u ‖ H 1 ( ω 1 ) + ‖ w ‖ H 1 ( ω 1 ) ) α ( ‖ u ‖ H 1 ( Ω 1 ) + ‖ w ‖ H 1 ( Ω 1 ) ) 1 − α + C k ‖ L u ‖ H − 1 ( Ω ) ≤ C ( ‖ u ‖ H 1 ( ω 1 ) + k ‖ L u ‖ H − 1 ( Ω ) ) α ( ‖ u ‖ H 1 ( Ω 1 ) + k ‖ L u ‖ H − 1 ( Ω ) ) 1 − α ‖ u ‖ H 1 ( B ) ≤ ‖ v ‖ H 1 ( B ) + ‖ w ‖ H 1 ( B ) ≤ C ‖ u ‖ H 1 ω 1 + ‖ w ‖ H 1 ω 1 α ‖ u ‖ H 1 Ω 1 + ‖ w ‖ H 1 Ω 1 1 − α + C k ‖ L u ‖ H − 1 ( Ω ) ≤ C ‖ u ‖ H 1 ω 1 + k ‖ L u ‖ H − 1 ( Ω ) α ‖ u ‖ H 1 Ω 1 + k ‖ L u ‖ H − 1 ( Ω ) 1 − α {:[||u||_(H^(1)(B)) <= ||v||_(H^(1)(B))+||w||_(H^(1)(B))],[ <= C(||u||_(H^(1)(omega_(1)))+||w||_(H^(1)(omega_(1))))^(alpha)(||u||_(H^(1)(Omega_(1)))+||w||_(H^(1)(Omega_(1))))^(1-alpha)+Ck||Lu||_(H^(-1)(Omega))],[ <= C(||u||_(H^(1)(omega_(1)))+k||Lu||_(H^(-1)(Omega)))^(alpha)(||u||_(H^(1)(Omega_(1)))+k||Lu||_(H^(-1)(Omega)))^(1-alpha)]:}\begin{aligned} \|u\|_{H^{1}(B)} & \leq\|v\|_{H^{1}(B)}+\|w\|_{H^{1}(B)} \\ & \leq C\left(\|u\|_{H^{1}\left(\omega_{1}\right)}+\|w\|_{H^{1}\left(\omega_{1}\right)}\right)^{\alpha}\left(\|u\|_{H^{1}\left(\Omega_{1}\right)}+\|w\|_{H^{1}\left(\Omega_{1}\right)}\right)^{1-\alpha}+C k\|\mathcal{L} u\|_{H^{-1}(\Omega)} \\ & \leq C\left(\|u\|_{H^{1}\left(\omega_{1}\right)}+k\|\mathcal{L} u\|_{H^{-1}(\Omega)}\right)^{\alpha}\left(\|u\|_{H^{1}\left(\Omega_{1}\right)}+k\|\mathcal{L} u\|_{H^{-1}(\Omega)}\right)^{1-\alpha} \end{aligned}‖u‖H1(B)≤‖v‖H1(B)+‖w‖H1(B)≤C(‖u‖H1(ω1)+‖w‖H1(ω1))α(‖u‖H1(Ω1)+‖w‖H1(Ω1))1−α+Ck‖Lu‖H−1(Ω)≤C(‖u‖H1(ω1)+k‖Lu‖H−1(Ω))α(‖u‖H1(Ω1)+k‖Lu‖H−1(Ω))1−α
Now we choose a cutoff function χ ∈ C 0 ∞ ( ω ) χ ∈ C 0 ∞ ( ω ) chi inC_(0)^(oo)(omega)\chi \in C_{0}^{\infty}(\omega)χ∈C0∞(ω) such that χ = 1 χ = 1 chi=1\chi=1χ=1 in ω 1 ω 1 omega_(1)\omega_{1}ω1 and χ u χ u chi u\chi uχu satisfies
L ( χ u ) = χ L u + [ L , χ ] u , ∂ n ( χ u ) + i k ( χ u ) = 0 on ∂ ω . L ( χ u ) = χ L u + [ L , χ ] u , ∂ n ( χ u ) + i k ( χ u ) = 0  on  ∂ ω . L(chi u)=chiLu+[L,chi]u,quaddel_(n)(chi u)+ik(chi u)=0" on "del omega.\mathcal{L}(\chi u)=\chi \mathcal{L} u+[\mathcal{L}, \chi] u, \quad \partial_{n}(\chi u)+i k(\chi u)=0 \text { on } \partial \omega .L(χu)=χLu+[L,χ]u,∂n(χu)+ik(χu)=0 on ∂ω.
Since the commutator [ L , χ ] [ L , χ ] [L,chi][\mathcal{L}, \chi][L,χ] is of first order, using again [3, Corollary 1.10] we obtain
‖ u ‖ H 1 ( ω 1 ) ≤ ‖ χ u ‖ H 1 ( ω ) ≤ C k ( ‖ [ L , χ ] u ‖ H − 1 ( ω ) + ‖ χ L u ‖ H − 1 ( ω ) ) ≤ C k ( ‖ u ‖ L 2 ( ω ) + ‖ L u ‖ H − 1 ( ω ) ) ‖ u ‖ H 1 ω 1 ≤ ‖ χ u ‖ H 1 ( ω ) ≤ C k ‖ [ L , χ ] u ‖ H − 1 ( ω ) + ‖ χ L u ‖ H − 1 ( ω ) ≤ C k ‖ u ‖ L 2 ( ω ) + ‖ L u ‖ H − 1 ( ω ) {:[||u||_(H^(1)(omega_(1))) <= ||chi u||_(H^(1)(omega)) <= Ck(||[L,chi]u||_(H^(-1)(omega))+||chiLu||_(H^(-1)(omega)))],[ <= Ck(||u||_(L^(2)(omega))+||Lu||_(H^(-1)(omega)))]:}\begin{aligned} \|u\|_{H^{1}\left(\omega_{1}\right)} & \leq\|\chi u\|_{H^{1}(\omega)} \leq C k\left(\|[\mathcal{L}, \chi] u\|_{H^{-1}(\omega)}+\|\chi \mathcal{L} u\|_{H^{-1}(\omega)}\right) \\ & \leq C k\left(\|u\|_{L^{2}(\omega)}+\|\mathcal{L} u\|_{H^{-1}(\omega)}\right) \end{aligned}‖u‖H1(ω1)≤‖χu‖H1(ω)≤Ck(‖[L,χ]u‖H−1(ω)+‖χLu‖H−1(ω))≤Ck(‖u‖L2(ω)+‖Lu‖H−1(ω))
The same argument for Ω 1 ⊂ Ω Ω 1 ⊂ Ω Omega_(1)sub Omega\Omega_{1} \subset \OmegaΩ1⊂Ω gives
‖ u ‖ H 1 ( Ω 1 ) ≤ C k ( ‖ u ‖ L 2 ( Ω ) + ‖ L u ‖ H − 1 ( Ω ) ) , ‖ u ‖ H 1 Ω 1 ≤ C k ‖ u ‖ L 2 ( Ω ) + ‖ L u ‖ H − 1 ( Ω ) , ||u||_(H^(1)(Omega_(1))) <= Ck(||u||_(L^(2)(Omega))+||Lu||_(H^(-1)(Omega))),\|u\|_{H^{1}\left(\Omega_{1}\right)} \leq C k\left(\|u\|_{L^{2}(\Omega)}+\|\mathcal{L} u\|_{H^{-1}(\Omega)}\right),‖u‖H1(Ω1)≤Ck(‖u‖L2(Ω)+‖Lu‖H−1(Ω)),
thus leading to the conclusion.
2.2. Shifted three-ball inequality. In this section we prove an estimate as in Corollary 2, but with the Sobolev indices shifted down one degree, and our starting point is again the Carleman estimate in Corollary 1. When shifting Carleman estimates, as we want to keep track of the large parameter τ τ tau\tauτ, it is convenient to use the semiclassical version of pseudodifferential calculus. We write ℏ > 0 ℏ > 0 ℏ > 0\hbar>0ℏ>0 for the semiclassical parameter that satisfies ℏ = 1 / τ ℏ = 1 / τ ℏ=1//tau\hbar=1 / \tauℏ=1/τ.
The semiclassical (pseudo)differential operators are (pseudo)differential operators where, roughly speaking, each derivative is multiplied by ℏ ℏ ℏ\hbarℏ, for the precise definition see Section 4.1 of [27]. The scale of semiclassical Bessel potentials is defined by
J s = ( 1 − ℏ 2 Δ ) s / 2 , s ∈ R J s = 1 − ℏ 2 Δ s / 2 , s ∈ R J^(s)=(1-ℏ^(2)Delta)^(s//2),quad s inRJ^{s}=\left(1-\hbar^{2} \Delta\right)^{s / 2}, \quad s \in \mathbb{R}Js=(1−ℏ2Δ)s/2,s∈R
and the semiclassical Sobolev spaces by
‖ u ‖ H scl s ( R n ) = ‖ J s u ‖ L 2 ( R n ) ‖ u ‖ H scl s R n = J s u L 2 R n ||u||_(H_(scl)^(s)(R^(n)))=||J^(s)u||_(L^(2)(R^(n)))\|u\|_{H_{\mathrm{scl}}^{s}\left(\mathbb{R}^{n}\right)}=\left\|J^{s} u\right\|_{L^{2}\left(\mathbb{R}^{n}\right)}‖u‖Hscls(Rn)=‖Jsu‖L2(Rn)
Then a semiclassical differential operator of order m m mmm is continuous from H scl m + s ( R n ) H scl m + s R n H_(scl)^(m+s)(R^(n))H_{\mathrm{scl}}^{m+s}\left(\mathbb{R}^{n}\right)Hsclm+s(Rn) to H scl s ( R n ) H scl s R n H_(scl)^(s)(R^(n))H_{\mathrm{scl}}^{s}\left(\mathbb{R}^{n}\right)Hscls(Rn), see e.g. Section 8.3 of [27].
We will give a shifting argument that is similar to that in Section 4 of [12]. To this end, we need the following pseudolocal and commutator estimates for semiclassical pseudodifferential operators, see e.g. (4.8) and (4.9) of [12]. Suppose that ψ , χ ∈ C 0 ∞ ( R n ) ψ , χ ∈ C 0 ∞ R n psi,chi inC_(0)^(oo)(R^(n))\psi, \chi \in C_{0}^{\infty}\left(\mathbb{R}^{n}\right)ψ,χ∈C0∞(Rn) and that χ = 1 χ = 1 chi=1\chi=1χ=1
near supp ( ψ ) supp ( ψ ) supp(psi)\operatorname{supp}(\psi)supp(ψ), and let A , B A , B A,BA, BA,B be two semiclassical pseudodifferential operators of orders s , m s , m s,ms, ms,m, respectively. Then for all p , q , N ∈ R p , q , N ∈ R p,q,N inRp, q, N \in \mathbb{R}p,q,N∈R, there is C > 0 C > 0 C > 0C>0C>0
(11) ‖ ( 1 − χ ) A ( ψ u ) ‖ H scl p ( R n ) ≤ C ℏ N ‖ u ‖ H scl q ( R n ) (12) ‖ [ A , B ] u ‖ H scl p ( R n ) ≤ C ℏ ‖ u ‖ H scl p + s + m − 1 ( R n ) (11) ‖ ( 1 − χ ) A ( ψ u ) ‖ H scl p R n ≤ C ℏ N ‖ u ‖ H scl q R n (12) ‖ [ A , B ] u ‖ H scl p R n ≤ C ℏ ‖ u ‖ H scl p + s + m − 1 R n {:[(11)||(1-chi)A(psi u)||_(H_(scl)^(p)(R^(n))) <= Cℏ^(N)||u||_(H_(scl)^(q)(R^(n)))],[(12)||[A","B]u||_(H_(scl)^(p)(R^(n))) <= Cℏ||u||_(H_(scl)^(p+s+m-1)(R^(n)))]:}\begin{align*} \|(1-\chi) A(\psi u)\|_{H_{\mathrm{scl}}^{p}\left(\mathbb{R}^{n}\right)} & \leq C \hbar^{N}\|u\|_{H_{\mathrm{scl}}^{q}\left(\mathbb{R}^{n}\right)} \tag{11}\\ \|[A, B] u\|_{H_{\mathrm{scl}}^{p}\left(\mathbb{R}^{n}\right)} & \leq C \hbar\|u\|_{H_{\mathrm{scl}}^{p+s+m-1}\left(\mathbb{R}^{n}\right)} \tag{12} \end{align*}(11)‖(1−χ)A(ψu)‖Hsclp(Rn)≤CℏN‖u‖Hsclq(Rn)(12)‖[A,B]u‖Hsclp(Rn)≤Cℏ‖u‖Hsclp+s+m−1(Rn)
Both these estimates follow from the composition calculus, see e.g. [27, Theorem 4.12].
Let ϕ ϕ phi\phiϕ be as in Corollary 1 and set ℓ = ϕ / ℏ ℓ = ϕ / ℏ ℓ=phi//ℏ\ell=\phi / \hbarℓ=ϕ/ℏ and σ = Δ ℓ σ = Δ ℓ sigma=Deltaℓ\sigma=\Delta \ellσ=Δℓ in Lemma 1. Then
( e ϕ / ℏ Δ e − ϕ / ℏ v + k 2 v ) 2 / 2 ≥ 2 ℏ − 1 D 2 ϕ ( ∇ v , ∇ v ) + 2 ℏ − 3 D 2 ϕ ( ∇ ϕ , ∇ ϕ ) v 2 + div ( b ∇ v + B ) + ℏ − 1 ( ∇ Δ ϕ , ∇ v ) v e ϕ / ℏ Δ e − ϕ / ℏ v + k 2 v 2 / 2 ≥ 2 ℏ − 1 D 2 ϕ ( ∇ v , ∇ v ) + 2 ℏ − 3 D 2 ϕ ( ∇ ϕ , ∇ ϕ ) v 2 + div ( b ∇ v + B ) + ℏ − 1 ( ∇ Δ ϕ , ∇ v ) v {:[(e^(phi//ℏ)Deltae^(-phi//ℏ)v+k^(2)v)^(2)//2 >= 2ℏ^(-1)D^(2)phi(grad v","grad v)+2ℏ^(-3)D^(2)phi(grad phi","grad phi)v^(2)],[+div(b grad v+B)+ℏ^(-1)(grad Delta phi","grad v)v]:}\begin{aligned} \left(e^{\phi / \hbar} \Delta e^{-\phi / \hbar} v+k^{2} v\right)^{2} / 2 \geq & 2 \hbar^{-1} D^{2} \phi(\nabla v, \nabla v)+2 \hbar^{-3} D^{2} \phi(\nabla \phi, \nabla \phi) v^{2} \\ & +\operatorname{div}(b \nabla v+B)+\hbar^{-1}(\nabla \Delta \phi, \nabla v) v \end{aligned}(eϕ/ℏΔe−ϕ/ℏv+k2v)2/2≥2ℏ−1D2ϕ(∇v,∇v)+2ℏ−3D2ϕ(∇ϕ,∇ϕ)v2+div(b∇v+B)+ℏ−1(∇Δϕ,∇v)v
Write P = e ϕ / ℏ ℏ 2 Δ e − ϕ / ℏ P = e ϕ / ℏ ℏ 2 Δ e − ϕ / ℏ P=e^(phi//ℏ)ℏ^(2)Deltae^(-phi//ℏ)P=e^{\phi / \hbar} \hbar^{2} \Delta e^{-\phi / \hbar}P=eϕ/ℏℏ2Δe−ϕ/ℏ and let v ∈ C 0 ∞ ( Ω ′ ) v ∈ C 0 ∞ Ω ′ v inC_(0)^(oo)(Omega^('))v \in C_{0}^{\infty}\left(\Omega^{\prime}\right)v∈C0∞(Ω′) where Ω ′ ⊂ R n Ω ′ ⊂ R n Omega^(')subR^(n)\Omega^{\prime} \subset \mathbb{R}^{n}Ω′⊂Rn is open and bounded, and Ω ¯ ⊂ Ω ′ Ω ¯ ⊂ Ω ′ bar(Omega)subOmega^(')\bar{\Omega} \subset \Omega^{\prime}Ω¯⊂Ω′. Then, rescaling by ℏ 4 ℏ 4 ℏ^(4)\hbar^{4}ℏ4,
C ‖ P v + ℏ 2 k 2 v ‖ L 2 ( R n ) 2 ≥ ℏ ‖ ℏ ∇ v ‖ L 2 ( R n ) 2 + ℏ ‖ v ‖ L 2 ( R n ) 2 − C ℏ 2 ‖ v ‖ H scl 1 ( R n ) 2 C P v + ℏ 2 k 2 v L 2 R n 2 ≥ ℏ ‖ ℏ ∇ v ‖ L 2 R n 2 + ℏ ‖ v ‖ L 2 R n 2 − C ℏ 2 ‖ v ‖ H scl 1 R n 2 C||Pv+ℏ^(2)k^(2)v||_(L^(2)(R^(n)))^(2) >= ℏ||ℏgrad v||_(L^(2)(R^(n)))^(2)+ℏ||v||_(L^(2)(R^(n)))^(2)-Cℏ^(2)||v||_(H_(scl)^(1)(R^(n)))^(2)C\left\|P v+\hbar^{2} k^{2} v\right\|_{L^{2}\left(\mathbb{R}^{n}\right)}^{2} \geq \hbar\|\hbar \nabla v\|_{L^{2}\left(\mathbb{R}^{n}\right)}^{2}+\hbar\|v\|_{L^{2}\left(\mathbb{R}^{n}\right)}^{2}-C \hbar^{2}\|v\|_{H_{\mathrm{scl}}^{1}\left(\mathbb{R}^{n}\right)}^{2}C‖Pv+ℏ2k2v‖L2(Rn)2≥ℏ‖ℏ∇v‖L2(Rn)2+ℏ‖v‖L2(Rn)2−Cℏ2‖v‖Hscl1(Rn)2
and for small enough ℏ > 0 ℏ > 0 ℏ > 0\hbar>0ℏ>0 we obtain
ℏ ‖ v ‖ H scl 1 ( R n ) ≤ C ‖ P v + ℏ 2 k 2 v ‖ L 2 ( R n ) ℏ ‖ v ‖ H scl 1 R n ≤ C P v + ℏ 2 k 2 v L 2 R n sqrtℏ||v||_(H_(scl)^(1)(R^(n))) <= C||Pv+ℏ^(2)k^(2)v||_(L^(2)(R^(n)))\sqrt{\hbar}\|v\|_{H_{\mathrm{scl}}^{1}\left(\mathbb{R}^{n}\right)} \leq C\left\|P v+\hbar^{2} k^{2} v\right\|_{L^{2}\left(\mathbb{R}^{n}\right)}ℏ‖v‖Hscl1(Rn)≤C‖Pv+ℏ2k2v‖L2(Rn)
Now the conjugated operator P P PPP is a semiclassical differential operator,
P u = e ϕ / ℏ ℏ 2 div ∇ ( e − ϕ / ℏ u ) = ℏ 2 Δ u − 2 ( ∇ ϕ , ℏ ∇ u ) − ℏ ( Δ ϕ ) u + | ∇ ϕ | 2 u P u = e ϕ / ℏ ℏ 2 div ∇ e − ϕ / ℏ u = ℏ 2 Δ u − 2 ( ∇ ϕ , ℏ ∇ u ) − ℏ ( Δ ϕ ) u + | ∇ ϕ | 2 u Pu=e^(phi//ℏ)ℏ^(2)div grad(e^(-phi//ℏ)u)=ℏ^(2)Delta u-2(grad phi,ℏgrad u)-ℏ(Delta phi)u+|grad phi|^(2)uP u=e^{\phi / \hbar} \hbar^{2} \operatorname{div} \nabla\left(e^{-\phi / \hbar} u\right)=\hbar^{2} \Delta u-2(\nabla \phi, \hbar \nabla u)-\hbar(\Delta \phi) u+|\nabla \phi|^{2} uPu=eϕ/ℏℏ2div∇(e−ϕ/ℏu)=ℏ2Δu−2(∇ϕ,ℏ∇u)−ℏ(Δϕ)u+|∇ϕ|2u
Let χ , ψ ∈ C 0 ∞ ( Ω ′ ) χ , ψ ∈ C 0 ∞ Ω ′ chi,psi inC_(0)^(oo)(Omega^('))\chi, \psi \in C_{0}^{\infty}\left(\Omega^{\prime}\right)χ,ψ∈C0∞(Ω′) and suppose that ψ = 1 ψ = 1 psi=1\psi=1ψ=1 near Ω Ω Omega\OmegaΩ and χ = 1 χ = 1 chi=1\chi=1χ=1 near supp ( ψ ) supp ( ψ ) supp(psi)\operatorname{supp}(\psi)supp(ψ). Then for v ∈ C 0 ∞ ( Ω ) v ∈ C 0 ∞ ( Ω ) v inC_(0)^(oo)(Omega)v \in C_{0}^{\infty}(\Omega)v∈C0∞(Ω),
‖ v ‖ H scl 1 + s ( R n ) ≤ ‖ χ J s v ‖ H scl 1 ( R n ) + ‖ ( 1 − χ ) J s ψ v ‖ H scl 1 ( R n ) ≤ C ‖ χ J s v ‖ H scl 1 ( R n ) ‖ v ‖ H scl 1 + s R n ≤ χ J s v H scl 1 R n + ( 1 − χ ) J s ψ v H scl 1 R n ≤ C χ J s v H scl 1 R n ||v||_(H_(scl)^(1+s)(R^(n))) <= ||chiJ^(s)v||_(H_(scl)^(1)(R^(n)))+||(1-chi)J^(s)psi v||_(H_(scl)^(1)(R^(n))) <= C||chiJ^(s)v||_(H_(scl)^(1)(R^(n)))\|v\|_{H_{\mathrm{scl}}^{1+s}\left(\mathbb{R}^{n}\right)} \leq\left\|\chi J^{s} v\right\|_{H_{\mathrm{scl}}^{1}\left(\mathbb{R}^{n}\right)}+\left\|(1-\chi) J^{s} \psi v\right\|_{H_{\mathrm{scl}}^{1}\left(\mathbb{R}^{n}\right)} \leq C\left\|\chi J^{s} v\right\|_{H_{\mathrm{scl}}^{1}\left(\mathbb{R}^{n}\right)}‖v‖Hscl1+s(Rn)≤‖χJsv‖Hscl1(Rn)+‖(1−χ)Jsψv‖Hscl1(Rn)≤C‖χJsv‖Hscl1(Rn)
where we used the pseudolocality (11) to absorb the second term on the right-hand side by the left-hand side. We have
(13) ℏ ‖ v ‖ H scl 1 + s ( R n ) ≤ C ℏ ‖ χ J s v ‖ H scl 1 ( R n ) ≤ C ‖ ( P + ℏ 2 k 2 ) χ J s v ‖ L 2 ( R n ) (13) ℏ ‖ v ‖ H scl 1 + s R n ≤ C ℏ χ J s v H scl 1 R n ≤ C P + ℏ 2 k 2 χ J s v L 2 R n {:(13)sqrtℏ||v||_(H_(scl)^(1+s)(R^(n))) <= Csqrtℏ||chiJ^(s)v||_(H_(scl)^(1)(R^(n))) <= C||(P+ℏ^(2)k^(2))chiJ^(s)v||_(L^(2)(R^(n))):}\begin{equation*} \sqrt{\hbar}\|v\|_{H_{\mathrm{scl}}^{1+s}\left(\mathbb{R}^{n}\right)} \leq C \sqrt{\hbar}\left\|\chi J^{s} v\right\|_{H_{\mathrm{scl}}^{1}\left(\mathbb{R}^{n}\right)} \leq C\left\|\left(P+\hbar^{2} k^{2}\right) \chi J^{s} v\right\|_{L^{2}\left(\mathbb{R}^{n}\right)} \tag{13} \end{equation*}(13)ℏ‖v‖Hscl1+s(Rn)≤Cℏ‖χJsv‖Hscl1(Rn)≤C‖(P+ℏ2k2)χJsv‖L2(Rn)
and using the commutator estimate (12), we have
‖ [ P , χ J s ] v ‖ L 2 ( R n ) ≤ C ℏ ‖ v ‖ H scl 1 + s ( R n ) P , χ J s v L 2 R n ≤ C ℏ ‖ v ‖ H scl 1 + s R n ||[P,chiJ^(s)]v||_(L^(2)(R^(n))) <= Cℏ||v||_(H_(scl)^(1+s)(R^(n)))\left\|\left[P, \chi J^{s}\right] v\right\|_{L^{2}\left(\mathbb{R}^{n}\right)} \leq C \hbar\|v\|_{H_{\mathrm{scl}}^{1+s}\left(\mathbb{R}^{n}\right)}‖[P,χJs]v‖L2(Rn)≤Cℏ‖v‖Hscl1+s(Rn)
This can be absorbed by the left-hand side of (13). Thus
ℏ ‖ v ‖ H scl 1 + s ( R n ) ≤ C ‖ χ J s ( P + ℏ 2 k 2 ) v ‖ L 2 ( R n ) ≤ C ‖ ( P + ℏ 2 k 2 ) v ‖ H scl s ( R n ) ℏ ‖ v ‖ H scl 1 + s R n ≤ C χ J s P + ℏ 2 k 2 v L 2 R n ≤ C P + ℏ 2 k 2 v H scl s R n sqrtℏ||v||_(H_(scl)^(1+s)(R^(n))) <= C||chiJ^(s)(P+ℏ^(2)k^(2))v||_(L^(2)(R^(n))) <= C||(P+ℏ^(2)k^(2))v||_(H_(scl)^(s)(R^(n)))\sqrt{\hbar}\|v\|_{H_{\mathrm{scl}}^{1+s}\left(\mathbb{R}^{n}\right)} \leq C\left\|\chi J^{s}\left(P+\hbar^{2} k^{2}\right) v\right\|_{L^{2}\left(\mathbb{R}^{n}\right)} \leq C\left\|\left(P+\hbar^{2} k^{2}\right) v\right\|_{H_{\mathrm{scl}}^{s}\left(\mathbb{R}^{n}\right)}ℏ‖v‖Hscl1+s(Rn)≤C‖χJs(P+ℏ2k2)v‖L2(Rn)≤C‖(P+ℏ2k2)v‖Hscls(Rn)
Take now s = − 1 s = − 1 s=-1s=-1s=−1 and let the cutoff χ χ chi\chiχ and the weight ϕ ϕ phi\phiϕ be as in the proof of Corollary 2, with the additional condition on χ χ chi\chiχ such that there is ψ ∈ C 0 ∞ ( B ( y , s ) ∪ ω ) ψ ∈ C 0 ∞ ( B ( y , s ) ∪ ω ) psi inC_(0)^(oo)(B(y,s)uu omega)\psi \in C_{0}^{\infty}(B(y, s) \cup \omega)ψ∈C0∞(B(y,s)∪ω) satisfying ψ = 1 ψ = 1 psi=1\psi=1ψ=1 in supp ( [ P , χ ] ) supp ( [ P , χ ] ) supp([P,chi])\operatorname{supp}([P, \chi])supp([P,χ]).
Let u ∈ C ∞ ( R n ) u ∈ C ∞ R n u inC^(oo)(R^(n))u \in C^{\infty}\left(\mathbb{R}^{n}\right)u∈C∞(Rn) and set w = e ϕ / ℏ u w = e ϕ / ℏ u w=e^(phi//ℏ)uw=e^{\phi / \hbar} uw=eϕ/ℏu. Then the previous estimate becomes
ℏ ‖ χ w ‖ L 2 ( R n ) ≤ C ‖ ( P + ℏ 2 k 2 ) χ w ‖ H scl − 1 ( R n ) ℏ ‖ χ w ‖ L 2 R n ≤ C P + ℏ 2 k 2 χ w H scl − 1 R n sqrtℏ||chi w||_(L^(2)(R^(n))) <= C||(P+ℏ^(2)k^(2))chi w||_(H_(scl)^(-1)(R^(n)))\sqrt{\hbar}\|\chi w\|_{L^{2}\left(\mathbb{R}^{n}\right)} \leq C\left\|\left(P+\hbar^{2} k^{2}\right) \chi w\right\|_{H_{\mathrm{scl}}^{-1}\left(\mathbb{R}^{n}\right)}ℏ‖χw‖L2(Rn)≤C‖(P+ℏ2k2)χw‖Hscl−1(Rn)
We have
‖ [ P , χ ] w ‖ H scl − 1 ( R n ) = ‖ [ P , χ ] ψ w ‖ H scl − 1 ( R n ) ≤ C ℏ ‖ ψ w ‖ L 2 ( R n ) ‖ [ P , χ ] w ‖ H scl − 1 R n = ‖ [ P , χ ] ψ w ‖ H scl − 1 R n ≤ C ℏ ‖ ψ w ‖ L 2 R n ||[P,chi]w||_(H_(scl)^(-1)(R^(n)))=||[P,chi]psi w||_(H_(scl)^(-1)(R^(n))) <= Cℏ||psi w||_(L^(2)(R^(n)))\|[P, \chi] w\|_{H_{\mathrm{scl}}^{-1}\left(\mathbb{R}^{n}\right)}=\|[P, \chi] \psi w\|_{H_{\mathrm{scl}}^{-1}\left(\mathbb{R}^{n}\right)} \leq C \hbar\|\psi w\|_{L^{2}\left(\mathbb{R}^{n}\right)}‖[P,χ]w‖Hscl−1(Rn)=‖[P,χ]ψw‖Hscl−1(Rn)≤Cℏ‖ψw‖L2(Rn)
Using the norm inequality ‖ ⋅ ‖ H scl − 1 ( R n ) ≤ C ℏ − 2 ‖ ⋅ ‖ H − 1 ( R n ) ‖ ⋅ ‖ H scl − 1 R n ≤ C ℏ − 2 ‖ ⋅ ‖ H − 1 R n ||*||_(H_(scl)^(-1)(R^(n))) <= Cℏ^(-2)||*||_(H^(-1)(R^(n)))\|\cdot\|_{H_{\mathrm{scl}}^{-1}\left(\mathbb{R}^{n}\right)} \leq C \hbar^{-2}\|\cdot\|_{H^{-1}\left(\mathbb{R}^{n}\right)}‖⋅‖Hscl−1(Rn)≤Cℏ−2‖⋅‖H−1(Rn), we thus obtain
ℏ ‖ χ e ϕ / ℏ u ‖ L 2 ( R n ) ≤ C ‖ χ ( e ϕ / ℏ Δ e − ϕ / ℏ + k 2 ) w ‖ H scl − 1 ( R n ) + C ℏ ‖ ψ w ‖ L 2 ( R n ) ≤ C ℏ − 2 ‖ χ e ϕ / ℏ ( Δ u + k 2 u ) ‖ H − 1 ( R n ) + C ℏ ‖ ψ e ϕ / ℏ u ‖ L 2 ( R n ) ℏ χ e ϕ / ℏ u L 2 R n ≤ C χ e ϕ / ℏ Δ e − ϕ / ℏ + k 2 w H scl − 1 R n + C ℏ ‖ ψ w ‖ L 2 R n ≤ C ℏ − 2 χ e ϕ / ℏ Δ u + k 2 u H − 1 R n + C ℏ ψ e ϕ / ℏ u L 2 R n {:[sqrtℏ||chie^(phi//ℏ)u||_(L^(2)(R^(n))) <= C||chi(e^(phi//ℏ)Deltae^(-phi//ℏ)+k^(2))w||_(H_(scl)^(-1)(R^(n)))+Cℏ||psi w||_(L^(2)(R^(n)))],[ <= Cℏ^(-2)||chie^(phi//ℏ)(Delta u+k^(2)u)||_(H^(-1)(R^(n)))+Cℏ||psie^(phi//ℏ)u||_(L^(2)(R^(n)))]:}\begin{aligned} \sqrt{\hbar}\left\|\chi e^{\phi / \hbar} u\right\|_{L^{2}\left(\mathbb{R}^{n}\right)} & \leq C\left\|\chi\left(e^{\phi / \hbar} \Delta e^{-\phi / \hbar}+k^{2}\right) w\right\|_{H_{\mathrm{scl}}^{-1}\left(\mathbb{R}^{n}\right)}+C \hbar\|\psi w\|_{L^{2}\left(\mathbb{R}^{n}\right)} \\ & \leq C \hbar^{-2}\left\|\chi e^{\phi / \hbar}\left(\Delta u+k^{2} u\right)\right\|_{H^{-1}\left(\mathbb{R}^{n}\right)}+C \hbar\left\|\psi e^{\phi / \hbar} u\right\|_{L^{2}\left(\mathbb{R}^{n}\right)} \end{aligned}ℏ‖χeϕ/ℏu‖L2(Rn)≤C‖χ(eϕ/ℏΔe−ϕ/ℏ+k2)w‖Hscl−1(Rn)+Cℏ‖ψw‖L2(Rn)≤Cℏ−2‖χeϕ/ℏ(Δu+k2u)‖H−1(Rn)+Cℏ‖ψeϕ/ℏu‖L2(Rn)
Using the same notation as in the proof of Corollary 2, due to the choice of ψ ψ psi\psiψ we get
e ρ 2 / ℏ ‖ u ‖ L 2 ( B ) ≤ C e ( β + R ) 2 / ℏ ( ℏ − 7 2 ‖ Δ u + k 2 u ‖ H − 1 ( Ω ) + ℏ 1 2 ‖ u ‖ L 2 ( ω ) ) + C e s 2 / ℏ ℏ 1 2 ‖ u ‖ L 2 ( Ω ) , e ρ 2 / ℏ ‖ u ‖ L 2 ( B ) ≤ C e ( β + R ) 2 / ℏ ℏ − 7 2 Δ u + k 2 u H − 1 ( Ω ) + ℏ 1 2 ‖ u ‖ L 2 ( ω ) + C e s 2 / ℏ ℏ 1 2 ‖ u ‖ L 2 ( Ω ) , e^(rho^(2)//ℏ)||u||_(L^(2)(B)) <= Ce^((beta+R)^(2)//ℏ)(ℏ^(-(7)/(2))||Delta u+k^(2)u||_(H^(-1)(Omega))+ℏ^((1)/(2))||u||_(L^(2)(omega)))+Ce^(s^(2)//ℏ)ℏ^((1)/(2))||u||_(L^(2)(Omega)),e^{\rho^{2} / \hbar}\|u\|_{L^{2}(B)} \leq C e^{(\beta+R)^{2} / \hbar}\left(\hbar^{-\frac{7}{2}}\left\|\Delta u+k^{2} u\right\|_{H^{-1}(\Omega)}+\hbar^{\frac{1}{2}}\|u\|_{L^{2}(\omega)}\right)+C e^{s^{2} / \hbar} \hbar^{\frac{1}{2}}\|u\|_{L^{2}(\Omega)},eρ2/ℏ‖u‖L2(B)≤Ce(β+R)2/ℏ(ℏ−72‖Δu+k2u‖H−1(Ω)+ℏ12‖u‖L2(ω))+Ces2/ℏℏ12‖u‖L2(Ω),
for small enough ℏ > 0 ℏ > 0 ℏ > 0\hbar>0ℏ>0. Absorbing the negative power of ℏ ℏ ℏ\hbarℏ in the exponential, and using [22, Lemma 5.2], we conclude the proof of the following result.
Lemma 2. Let ω ⊂ B ⊂ Ω ω ⊂ B ⊂ Ω omega sub B sub Omega\omega \subset B \subset \Omegaω⊂B⊂Ω be defined as in Corollary 2. Then there are C > 0 C > 0 C > 0C>0C>0 and α ∈ ( 0 , 1 ) α ∈ ( 0 , 1 ) alpha in(0,1)\alpha \in(0,1)α∈(0,1) such that
‖ u ‖ L 2 ( B ) ≤ C ( ‖ u ‖ L 2 ( ω ) + ‖ Δ u + k 2 u ‖ H − 1 ( Ω ) ) α ‖ u ‖ L 2 ( Ω ) 1 − α ‖ u ‖ L 2 ( B ) ≤ C ‖ u ‖ L 2 ( ω ) + Δ u + k 2 u H − 1 ( Ω ) α ‖ u ‖ L 2 ( Ω ) 1 − α ||u||_(L^(2)(B)) <= C(||u||_(L^(2)(omega))+||Delta u+k^(2)u||_(H^(-1)(Omega)))^(alpha)||u||_(L^(2)(Omega))^(1-alpha)\|u\|_{L^{2}(B)} \leq C\left(\|u\|_{L^{2}(\omega)}+\left\|\Delta u+k^{2} u\right\|_{H^{-1}(\Omega)}\right)^{\alpha}\|u\|_{L^{2}(\Omega)}^{1-\alpha}‖u‖L2(B)≤C(‖u‖L2(ω)+‖Δu+k2u‖H−1(Ω))α‖u‖L2(Ω)1−α

3. Stabilized finite element method

We aim to solve the unique continuation problem for the Helmholtz equation
(14) Δ u + k 2 u = − f in Ω , u = q | ω , (14) Δ u + k 2 u = − f  in  Ω , u = q ω , {:(14)Delta u+k^(2)u=-f" in "Omega","quad u=q|_(omega)",":}\begin{equation*} \Delta u+k^{2} u=-f \text { in } \Omega, \quad u=\left.q\right|_{\omega}, \tag{14} \end{equation*}(14)Δu+k2u=−f in Ω,u=q|ω,
where ω ⊂ Ω ⊂ R 1 + n ω ⊂ Ω ⊂ R 1 + n omega sub Omega subR^(1+n)\omega \subset \Omega \subset \mathbb{R}^{1+n}ω⊂Ω⊂R1+n are open, f ∈ H − 1 ( Ω ) f ∈ H − 1 ( Ω ) f inH^(-1)(Omega)f \in H^{-1}(\Omega)f∈H−1(Ω) and q ∈ L 2 ( ω ) q ∈ L 2 ( ω ) q inL^(2)(omega)q \in L^{2}(\omega)q∈L2(ω) are given. Following the optimization based approach in [5, 8] we will make use of the continuum stability estimates in Section 2 when deriving error estimates for the finite element approximation.
3.1. Discretization. Consider a family T = { T h } h > 0 T = T h h > 0 T={T_(h)}_(h > 0)\mathcal{T}=\left\{\mathcal{T}_{h}\right\}_{h>0}T={Th}h>0 of triangulations of Ω Ω Omega\OmegaΩ consisting of simplices such that the intersection of any two distinct ones is either a common vertex, a common edge or a common face. Also, assume that the family T T T\mathcal{T}T is quasi-uniform. Let
V h = { u ∈ C ( Ω ¯ ) : u | K ∈ P 1 ( K ) , K ∈ T h } V h = u ∈ C ( Ω ¯ ) : u K ∈ P 1 ( K ) , K ∈ T h V_(h)={u in C(( bar(Omega))):u|_(K)inP_(1)(K),K inT_(h)}V_{h}=\left\{u \in C(\bar{\Omega}):\left.u\right|_{K} \in \mathbb{P}_{1}(K), K \in \mathcal{T}_{h}\right\}Vh={u∈C(Ω¯):u|K∈P1(K),K∈Th}
be the H 1 H 1 H^(1)H^{1}H1-conformal approximation space based on the P 1 P 1 P_(1)\mathbb{P}_{1}P1 finite element and let
W h = V h ∩ H 0 1 ( Ω ) W h = V h ∩ H 0 1 ( Ω ) W_(h)=V_(h)nnH_(0)^(1)(Omega)W_{h}=V_{h} \cap H_{0}^{1}(\Omega)Wh=Vh∩H01(Ω)
Consider the orthogonal L 2 L 2 L^(2)L^{2}L2-projection Π h : L 2 ( Ω ) → V h Π h : L 2 ( Ω ) → V h Pi_(h):L^(2)(Omega)rarrV_(h)\Pi_{h}: L^{2}(\Omega) \rightarrow V_{h}Πh:L2(Ω)→Vh, which satisfies
( u − Π h u , v ) L 2 ( Ω ) = 0 , u ∈ L 2 ( Ω ) , v ∈ V h ‖ Π h u ‖ L 2 ( Ω ) ≤ ‖ u ‖ L 2 ( Ω ) , u ∈ L 2 ( Ω ) u − Π h u , v L 2 ( Ω ) = 0 , u ∈ L 2 ( Ω ) , v ∈ V h Π h u L 2 ( Ω ) ≤ ‖ u ‖ L 2 ( Ω ) , u ∈ L 2 ( Ω ) {:[(u-Pi_(h)u,v)_(L^(2)(Omega))=0","quad u inL^(2)(Omega)","v inV_(h)],[||Pi_(h)u||_(L^(2)(Omega)) <= ||u||_(L^(2)(Omega))","quad u inL^(2)(Omega)]:}\begin{aligned} \left(u-\Pi_{h} u, v\right)_{L^{2}(\Omega)} & =0, \quad u \in L^{2}(\Omega), v \in V_{h} \\ \left\|\Pi_{h} u\right\|_{L^{2}(\Omega)} & \leq\|u\|_{L^{2}(\Omega)}, \quad u \in L^{2}(\Omega) \end{aligned}(u−Πhu,v)L2(Ω)=0,u∈L2(Ω),v∈Vh‖Πhu‖L2(Ω)≤‖u‖L2(Ω),u∈L2(Ω)
and the Scott-Zhang interpolator π h : H 1 ( Ω ) → V h π h : H 1 ( Ω ) → V h pi_(h):H^(1)(Omega)rarrV_(h)\pi_{h}: H^{1}(\Omega) \rightarrow V_{h}πh:H1(Ω)→Vh, that preserves vanishing Dirichlet boundary conditions. Both operators have the following stability and approximation properties, see e.g. [13, Chapter 1],
(15) ‖ i h u ‖ H 1 ( Ω ) ≤ C ‖ u ‖ H 1 ( Ω ) , u ∈ H 1 ( Ω ) , (16) ‖ u − i h u ‖ H m ( Ω ) ≤ C h k − m ‖ u ‖ H k ( Ω ) , u ∈ H k ( Ω ) , (15) i h u H 1 ( Ω ) ≤ C ‖ u ‖ H 1 ( Ω ) , u ∈ H 1 ( Ω ) , (16) u − i h u H m ( Ω ) ≤ C h k − m ‖ u ‖ H k ( Ω ) , u ∈ H k ( Ω ) , {:[(15)||i_(h)u||_(H^(1)(Omega)) <= C||u||_(H^(1)(Omega))","u inH^(1)(Omega)","],[(16)||u-i_(h)u||_(H^(m)(Omega)) <= Ch^(k-m)||u||_(H^(k)(Omega))","u inH^(k)(Omega)","]:}\begin{align*} \left\|i_{h} u\right\|_{H^{1}(\Omega)} & \leq C\|u\|_{H^{1}(\Omega)}, & & u \in H^{1}(\Omega), \tag{15}\\ \left\|u-i_{h} u\right\|_{H^{m}(\Omega)} & \leq C h^{k-m}\|u\|_{H^{k}(\Omega)}, & & u \in H^{k}(\Omega), \tag{16} \end{align*}(15)‖ihu‖H1(Ω)≤C‖u‖H1(Ω),u∈H1(Ω),(16)‖u−ihu‖Hm(Ω)≤Chk−m‖u‖Hk(Ω),u∈Hk(Ω),
where i = π , Π , k = 1 , 2 i = π , Π , k = 1 , 2 i=pi,Pi,k=1,2i=\pi, \Pi, k=1,2i=π,Π,k=1,2 and m = 0 , k − 1 m = 0 , k − 1 m=0,k-1m=0, k-1m=0,k−1.
The regularization on the discrete level will be based on the L 2 L 2 L^(2)L^{2}L2-control of the gradient jumps over elements edges using the jump stabilizer
J ( u , u ) = ∑ F ∈ F h ∫ F h [ [ n ⋅ ∇ u ] ] 2 d s , u ∈ V h J ( u , u ) = ∑ F ∈ F h   ∫ F   h [ [ n ⋅ ∇ u ] ] 2 d s , u ∈ V h J(u,u)=sum_(F inF_(h))int_(F)h[[n*grad u]]^(2)ds,quad u inV_(h)\mathcal{J}(u, u)=\sum_{F \in \mathcal{F}_{h}} \int_{F} h \llbracket n \cdot \nabla u \rrbracket^{2} d s, \quad u \in V_{h}J(u,u)=∑F∈Fh∫Fh[[n⋅∇u]]2ds,u∈Vh
where F h F h F_(h)\mathcal{F}_{h}Fh is the set of all internal faces, and the jump over F ∈ F h F ∈ F h F inF_(h)F \in \mathcal{F}_{h}F∈Fh is given by
[ [ n ⋅ ∇ u ] ] F = n 1 ⋅ ∇ u | K 1 + n 2 ⋅ ∇ u | K 2 , [ [ n ⋅ ∇ u ] ] F = n 1 ⋅ ∇ u K 1 + n 2 ⋅ ∇ u K 2 , [[n*grad u]]_(F)=n_(1)*grad u|_(K_(1))+n_(2)*grad u|_(K_(2)),\llbracket n \cdot \nabla u \rrbracket_{F}=\left.n_{1} \cdot \nabla u\right|_{K_{1}}+\left.n_{2} \cdot \nabla u\right|_{K_{2}},[[n⋅∇u]]F=n1⋅∇u|K1+n2⋅∇u|K2,
with K 1 , K 2 ∈ T h K 1 , K 2 ∈ T h K_(1),K_(2)inT_(h)K_{1}, K_{2} \in \mathcal{T}_{h}K1,K2∈Th being two simplices such that K 1 ∩ K 2 = F K 1 ∩ K 2 = F K_(1)nnK_(2)=FK_{1} \cap K_{2}=FK1∩K2=F, and n j n j n_(j)n_{j}nj the outward normal of K j , j = 1 , 2 K j , j = 1 , 2 K_(j),j=1,2K_{j}, j=1,2Kj,j=1,2. The face subscript is omitted when there is no ambiguity.
Lemma 3. There is C > 0 C > 0 C > 0C>0C>0 such that all u ∈ V h , v ∈ H 0 1 ( Ω ) , w ∈ H 2 ( Ω ) u ∈ V h , v ∈ H 0 1 ( Ω ) , w ∈ H 2 ( Ω ) u inV_(h),v inH_(0)^(1)(Omega),w inH^(2)(Omega)u \in V_{h}, v \in H_{0}^{1}(\Omega), w \in H^{2}(\Omega)u∈Vh,v∈H01(Ω),w∈H2(Ω) and h > 0 h > 0 h > 0h>0h>0 satisfy
(17) ( ∇ u , ∇ v ) L 2 ( Ω ) ≤ C J ( u , u ) 1 / 2 ( h − 1 ‖ v ‖ L 2 ( Ω ) + ‖ v ‖ H 1 ( Ω ) ) (18) J ( i h w , i h w ) ≤ C h 2 ‖ w ‖ H 2 ( Ω ) 2 , i ∈ { π , Π } . (17) ( ∇ u , ∇ v ) L 2 ( Ω ) ≤ C J ( u , u ) 1 / 2 h − 1 ‖ v ‖ L 2 ( Ω ) + ‖ v ‖ H 1 ( Ω ) (18) J i h w , i h w ≤ C h 2 ‖ w ‖ H 2 ( Ω ) 2 , i ∈ { π , Π } . {:[(17)(grad u","grad v)_(L^(2)(Omega)) <= CJ(u","u)^(1//2)(h^(-1)||v||_(L^(2)(Omega))+||v||_(H^(1)(Omega)))],[(18)J(i_(h)w,i_(h)w) <= Ch^(2)||w||_(H^(2)(Omega))^(2)","quad i in{pi","Pi}.]:}\begin{align*} (\nabla u, \nabla v)_{L^{2}(\Omega)} & \leq C \mathcal{J}(u, u)^{1 / 2}\left(h^{-1}\|v\|_{L^{2}(\Omega)}+\|v\|_{H^{1}(\Omega)}\right) \tag{17}\\ \mathcal{J}\left(i_{h} w, i_{h} w\right) & \leq C h^{2}\|w\|_{H^{2}(\Omega)}^{2}, \quad i \in\{\pi, \Pi\} . \tag{18} \end{align*}(17)(∇u,∇v)L2(Ω)≤CJ(u,u)1/2(h−1‖v‖L2(Ω)+‖v‖H1(Ω))(18)J(ihw,ihw)≤Ch2‖w‖H2(Ω)2,i∈{π,Π}.
Proof. See [10, Lemma 2] when the interpolator is π h π h pi_(h)\pi_{h}πh. Since this proof uses just the approximation properties of π h π h pi_(h)\pi_{h}πh, it holds verbatim for Π h Π h Pi_(h)\Pi_{h}Πh.
Adopting the notation
a ( u , z ) = ( ∇ u , ∇ z ) L 2 ( Ω ) , G f ( u , z ) = a ( u , z ) − k 2 ( u , z ) L 2 ( Ω ) − ⟨ f , z ⟩ , G = G 0 , a ( u , z ) = ( ∇ u , ∇ z ) L 2 ( Ω ) , G f ( u , z ) = a ( u , z ) − k 2 ( u , z ) L 2 ( Ω ) − ⟨ f , z ⟩ , G = G 0 , a(u,z)=(grad u,grad z)_(L^(2)(Omega)),quadG_(f)(u,z)=a(u,z)-k^(2)(u,z)_(L^(2)(Omega))-(:f,z:),quad G=G_(0),a(u, z)=(\nabla u, \nabla z)_{L^{2}(\Omega)}, \quad G_{f}(u, z)=a(u, z)-k^{2}(u, z)_{L^{2}(\Omega)}-\langle f, z\rangle, \quad G=G_{0},a(u,z)=(∇u,∇z)L2(Ω),Gf(u,z)=a(u,z)−k2(u,z)L2(Ω)−⟨f,z⟩,G=G0,
we write for u ∈ H 1 ( Ω ) u ∈ H 1 ( Ω ) u inH^(1)(Omega)u \in H^{1}(\Omega)u∈H1(Ω) the weak formulation of Δ u + k 2 u = − f Δ u + k 2 u = − f Delta u+k^(2)u=-f\Delta u+k^{2} u=-fΔu+k2u=−f as
G f ( u , z ) = 0 , z ∈ H 0 1 ( Ω ) G f ( u , z ) = 0 , z ∈ H 0 1 ( Ω ) G_(f)(u,z)=0,quad z inH_(0)^(1)(Omega)G_{f}(u, z)=0, \quad z \in H_{0}^{1}(\Omega)Gf(u,z)=0,z∈H01(Ω)
Our approach is to find the saddle points of the Lagrangian functional
L q , f ( u , z ) = 1 2 ‖ u − q ‖ ω 2 + 1 2 s ( u , u ) − 1 2 s ∗ ( z , z ) + G f ( u , z ) , L q , f ( u , z ) = 1 2 ‖ u − q ‖ ω 2 + 1 2 s ( u , u ) − 1 2 s ∗ ( z , z ) + G f ( u , z ) , L_(q,f)(u,z)=(1)/(2)||u-q||_(omega)^(2)+(1)/(2)s(u,u)-(1)/(2)s^(**)(z,z)+G_(f)(u,z),L_{q, f}(u, z)=\frac{1}{2}\|u-q\|_{\omega}^{2}+\frac{1}{2} s(u, u)-\frac{1}{2} s^{*}(z, z)+G_{f}(u, z),Lq,f(u,z)=12‖u−q‖ω2+12s(u,u)−12s∗(z,z)+Gf(u,z),
where ‖ ⋅ ‖ ω ‖ ⋅ ‖ ω ||*||_(omega)\|\cdot\|_{\omega}‖⋅‖ω denotes ‖ ⋅ ‖ L 2 ( ω ) ‖ ⋅ ‖ L 2 ( ω ) ||*||_(L^(2)(omega))\|\cdot\|_{L^{2}(\omega)}‖⋅‖L2(ω), and s s sss and s ∗ s ∗ s^(**)s^{*}s∗ are stabilizing (regularizing) terms for the primal and dual variables that should be consistent and vanish at optimal rates. The stabilization must control certain residual quantities representing the data of the error equation. The primal stabilizer will be based on the continuous interior penalty given by J J J\mathcal{J}J. It must take into account the zeroth order term of the Helmholtz operator. The dual variable can be stabilized in the H 1 H 1 H^(1)H^{1}H1-seminorm. Notice that when the PDE-constraint is satisfied, z = 0 z = 0 z=0z=0z=0 is the solution for the dual variable of the saddle point, thus the stabilizer s ∗ s ∗ s^(**)s^{*}s∗ is consistent. Hence we make the following choice
s ( u , u ) = J ( u , u ) + ‖ h k 2 u ‖ L 2 ( Ω ) 2 , s ∗ = a . s ( u , u ) = J ( u , u ) + h k 2 u L 2 ( Ω ) 2 , s ∗ = a . s(u,u)=J(u,u)+||hk^(2)u||_(L^(2)(Omega))^(2),quads^(**)=a.s(u, u)=\mathcal{J}(u, u)+\left\|h k^{2} u\right\|_{L^{2}(\Omega)}^{2}, \quad s^{*}=a .s(u,u)=J(u,u)+‖hk2u‖L2(Ω)2,s∗=a.
For a detailed presentation of such discrete stabilizing operators we refer the reader to [5] or [7]. We define on V h V h V_(h)V_{h}Vh and W h W h W_(h)W_{h}Wh, respectively, the norms
‖ u ‖ V = s ( u , u ) 1 / 2 , u ∈ V h , ‖ z ‖ W = s ∗ ( z , z ) 1 / 2 , z ∈ W h ‖ u ‖ V = s ( u , u ) 1 / 2 , u ∈ V h , ‖ z ‖ W = s ∗ ( z , z ) 1 / 2 , z ∈ W h ||u||_(V)=s(u,u)^(1//2),quad u inV_(h),quad||z||_(W)=s^(**)(z,z)^(1//2),quad z inW_(h)\|u\|_{V}=s(u, u)^{1 / 2}, \quad u \in V_{h}, \quad\|z\|_{W}=s^{*}(z, z)^{1 / 2}, \quad z \in W_{h}‖u‖V=s(u,u)1/2,u∈Vh,‖z‖W=s∗(z,z)1/2,z∈Wh
together with the norm on V h × W h V h × W h V_(h)xxW_(h)V_{h} \times W_{h}Vh×Wh defined by
‖ ( u , z ) ‖ 2 = ‖ u ‖ V 2 + ‖ u ‖ ω 2 + ‖ z ‖ W 2 ‖ ( u , z ) ‖ 2 = ‖ u ‖ V 2 + ‖ u ‖ ω 2 + ‖ z ‖ W 2 ||(u,z)||^(2)=||u||_(V)^(2)+||u||_(omega)^(2)+||z||_(W)^(2)\|(u, z)\|^{2}=\|u\|_{V}^{2}+\|u\|_{\omega}^{2}+\|z\|_{W}^{2}‖(u,z)‖2=‖u‖V2+‖u‖ω2+‖z‖W2
The saddle points ( u , z ) ∈ V h × W h ( u , z ) ∈ V h × W h (u,z)inV_(h)xxW_(h)(u, z) \in V_{h} \times W_{h}(u,z)∈Vh×Wh of the Lagrangian L q , f L q , f L_(q,f)L_{q, f}Lq,f satisfy
(19) A [ ( u , z ) , ( v , w ) ] = ( q , v ) ω + ⟨ f , w ⟩ , ( v , w ) ∈ V h × W h (19) A [ ( u , z ) , ( v , w ) ] = ( q , v ) ω + ⟨ f , w ⟩ , ( v , w ) ∈ V h × W h {:(19)A[(u","z)","(v","w)]=(q","v)_(omega)+(:f","w:)","quad(v","w)inV_(h)xxW_(h):}\begin{equation*} A[(u, z),(v, w)]=(q, v)_{\omega}+\langle f, w\rangle, \quad(v, w) \in V_{h} \times W_{h} \tag{19} \end{equation*}(19)A[(u,z),(v,w)]=(q,v)ω+⟨f,w⟩,(v,w)∈Vh×Wh
where A A AAA is the symmetric bilinear form
A [ ( u , z ) , ( v , w ) ] = ( u , v ) ω + s ( u , v ) + G ( v , z ) − s ∗ ( z , w ) + G ( u , w ) A [ ( u , z ) , ( v , w ) ] = ( u , v ) ω + s ( u , v ) + G ( v , z ) − s ∗ ( z , w ) + G ( u , w ) A[(u,z),(v,w)]=(u,v)_(omega)+s(u,v)+G(v,z)-s^(**)(z,w)+G(u,w)A[(u, z),(v, w)]=(u, v)_{\omega}+s(u, v)+G(v, z)-s^{*}(z, w)+G(u, w)A[(u,z),(v,w)]=(u,v)ω+s(u,v)+G(v,z)−s∗(z,w)+G(u,w)
Since A [ ( u , z ) , ( u , − z ) ] = ‖ u ‖ ω 2 + ‖ u ‖ V 2 + ‖ z ‖ W 2 A [ ( u , z ) , ( u , − z ) ] = ‖ u ‖ ω 2 + ‖ u ‖ V 2 + ‖ z ‖ W 2 A[(u,z),(u,-z)]=||u||_(omega)^(2)+||u||_(V)^(2)+||z||_(W)^(2)A[(u, z),(u,-z)]=\|u\|_{\omega}^{2}+\|u\|_{V}^{2}+\|z\|_{W}^{2}A[(u,z),(u,−z)]=‖u‖ω2+‖u‖V2+‖z‖W2 we have the following inf-sup condition
(20) sup ( v , w ) ∈ V h × W h A [ ( u , z ) , ( v , w ) ] ‖ ( v , w ) ‖ ‖ ≥ ‖ ( u , z ) ‖ (20) sup ( v , w ) ∈ V h × W h   A [ ( u , z ) , ( v , w ) ] ‖ ( v , w ) ‖ ‖ ≥ ‖ ( u , z ) ‖ {:(20)s u p_((v,w)inV_(h)xxW_(h))(A[(u,z),(v,w)])/(||(v,w)||||) >= ||(u","z)||:}\begin{equation*} \sup _{(v, w) \in V_{h} \times W_{h}} \frac{A[(u, z),(v, w)]}{\|(v, w)\| \|} \geq\|(u, z)\| \tag{20} \end{equation*}(20)sup(v,w)∈Vh×WhA[(u,z),(v,w)]‖(v,w)‖‖≥‖(u,z)‖
that guarantees a unique solution in V h × W h V h × W h V_(h)xxW_(h)V_{h} \times W_{h}Vh×Wh for (19).
3.2. Error estimates. We start by deriving some lower and upper bounds for the norm ‖ ⋅ ‖ V ‖ ⋅ ‖ V ||*||_(V)\|\cdot\|_{V}‖⋅‖V. For u h ∈ V h , z ∈ H 0 1 ( Ω ) u h ∈ V h , z ∈ H 0 1 ( Ω ) u_(h)inV_(h),z inH_(0)^(1)(Omega)u_{h} \in V_{h}, z \in H_{0}^{1}(\Omega)uh∈Vh,z∈H01(Ω), we use (17) to bound
G ( u h , z ) = ( ∇ u h , ∇ z ) L 2 ( Ω ) − k 2 ( u h , z ) L 2 ( Ω ) ≤ C J ( u h , u h ) 1 / 2 ( h − 1 ‖ z ‖ L 2 ( Ω ) + ‖ z ‖ H 1 ( Ω ) ) + k 2 ‖ u h ‖ L 2 ( Ω ) ‖ z ‖ L 2 ( Ω ) G u h , z = ∇ u h , ∇ z L 2 ( Ω ) − k 2 u h , z L 2 ( Ω ) ≤ C J u h , u h 1 / 2 h − 1 ‖ z ‖ L 2 ( Ω ) + ‖ z ‖ H 1 ( Ω ) + k 2 u h L 2 ( Ω ) ‖ z ‖ L 2 ( Ω ) {:[G(u_(h),z)=(gradu_(h),grad z)_(L^(2)(Omega))-k^(2)(u_(h),z)_(L^(2)(Omega))],[ <= CJ(u_(h),u_(h))^(1//2)(h^(-1)||z||_(L^(2)(Omega))+||z||_(H^(1)(Omega)))+k^(2)||u_(h)||_(L^(2)(Omega))||z||_(L^(2)(Omega))]:}\begin{aligned} G\left(u_{h}, z\right) & =\left(\nabla u_{h}, \nabla z\right)_{L^{2}(\Omega)}-k^{2}\left(u_{h}, z\right)_{L^{2}(\Omega)} \\ & \leq C \mathcal{J}\left(u_{h}, u_{h}\right)^{1 / 2}\left(h^{-1}\|z\|_{L^{2}(\Omega)}+\|z\|_{H^{1}(\Omega)}\right)+k^{2}\left\|u_{h}\right\|_{L^{2}(\Omega)}\|z\|_{L^{2}(\Omega)} \end{aligned}G(uh,z)=(∇uh,∇z)L2(Ω)−k2(uh,z)L2(Ω)≤CJ(uh,uh)1/2(h−1‖z‖L2(Ω)+‖z‖H1(Ω))+k2‖uh‖L2(Ω)‖z‖L2(Ω)
and hence
(21) G ( u h , z ) ≤ C ‖ u h ‖ V ( h − 1 ‖ z ‖ L 2 ( Ω ) + ‖ z ‖ H 1 ( Ω ) ) . (21) G u h , z ≤ C u h V h − 1 ‖ z ‖ L 2 ( Ω ) + ‖ z ‖ H 1 ( Ω ) . {:(21)G(u_(h),z) <= C||u_(h)||_(V)(h^(-1)||z||_(L^(2)(Omega))+||z||_(H^(1)(Omega))).:}\begin{equation*} G\left(u_{h}, z\right) \leq C\left\|u_{h}\right\|_{V}\left(h^{-1}\|z\|_{L^{2}(\Omega)}+\|z\|_{H^{1}(\Omega)}\right) . \tag{21} \end{equation*}(21)G(uh,z)≤C‖uh‖V(h−1‖z‖L2(Ω)+‖z‖H1(Ω)).
For u ∈ H 2 ( Ω ) u ∈ H 2 ( Ω ) u inH^(2)(Omega)u \in H^{2}(\Omega)u∈H2(Ω), from (18) and the stability of the L 2 L 2 L^(2)L^{2}L2-projection
‖ Π h u ‖ V 2 = J ( Π h u , Π h u ) + ‖ h k 2 Π h u ‖ L 2 ( Ω ) 2 ≤ C ( h 2 ‖ u ‖ H 2 ( Ω ) 2 + ‖ h k 2 u ‖ L 2 ( Ω ) 2 ) Π h u V 2 = J Π h u , Π h u + h k 2 Π h u L 2 ( Ω ) 2 ≤ C h 2 ‖ u ‖ H 2 ( Ω ) 2 + h k 2 u L 2 ( Ω ) 2 ||Pi_(h)u||_(V)^(2)=J(Pi_(h)u,Pi_(h)u)+||hk^(2)Pi_(h)u||_(L^(2)(Omega))^(2) <= C(h^(2)||u||_(H^(2)(Omega))^(2)+||hk^(2)u||_(L^(2)(Omega))^(2))\left\|\Pi_{h} u\right\|_{V}^{2}=\mathcal{J}\left(\Pi_{h} u, \Pi_{h} u\right)+\left\|h k^{2} \Pi_{h} u\right\|_{L^{2}(\Omega)}^{2} \leq C\left(h^{2}\|u\|_{H^{2}(\Omega)}^{2}+\left\|h k^{2} u\right\|_{L^{2}(\Omega)}^{2}\right)‖Πhu‖V2=J(Πhu,Πhu)+‖hk2Πhu‖L2(Ω)2≤C(h2‖u‖H2(Ω)2+‖hk2u‖L2(Ω)2)
implies
(22) ‖ Π h u ‖ V ≤ C h ( ‖ u ‖ H 2 ( Ω ) + k 2 ‖ u ‖ L 2 ( Ω ) ) = C h ‖ u ‖ ∗ , (22) Π h u V ≤ C h ‖ u ‖ H 2 ( Ω ) + k 2 ‖ u ‖ L 2 ( Ω ) = C h ‖ u ‖ ∗ , {:(22)||Pi_(h)u||_(V) <= Ch(||u||_(H^(2)(Omega))+k^(2)||u||_(L^(2)(Omega)))=Ch||u||_(**)",":}\begin{equation*} \left\|\Pi_{h} u\right\|_{V} \leq C h\left(\|u\|_{H^{2}(\Omega)}+k^{2}\|u\|_{L^{2}(\Omega)}\right)=C h\|u\|_{*}, \tag{22} \end{equation*}(22)‖Πhu‖V≤Ch(‖u‖H2(Ω)+k2‖u‖L2(Ω))=Ch‖u‖∗,
where ‖ u ‖ ∗ ‖ u ‖ ∗ ||u||_(**)\|u\|_{*}‖u‖∗ is defined as in (6).
Lemma 4. Let u ∈ H 2 ( Ω ) u ∈ H 2 ( Ω ) u inH^(2)(Omega)u \in H^{2}(\Omega)u∈H2(Ω) be the solution to (14) and ( u h , z h ) ∈ V h × W h u h , z h ∈ V h × W h (u_(h),z_(h))inV_(h)xxW_(h)\left(u_{h}, z_{h}\right) \in V_{h} \times W_{h}(uh,zh)∈Vh×Wh be the solution to (19). Then there exists C > 0 C > 0 C > 0C>0C>0 such that for all h ∈ ( 0 , 1 ) h ∈ ( 0 , 1 ) h in(0,1)h \in(0,1)h∈(0,1)
‖ ∣ ( u h − Π h u , z h ) ‖ ≤ C h ‖ u ‖ ∗ ∣ u h − Π h u , z h ≤ C h ‖ u ‖ ∗ ||∣(u_(h)-Pi_(h)u,z_(h))|| <= Ch||u||_(**)\left\|\mid\left(u_{h}-\Pi_{h} u, z_{h}\right)\right\| \leq C h\|u\|_{*}‖∣(uh−Πhu,zh)‖≤Ch‖u‖∗
Proof. Due to the inf-sup condition (20) it is enough to prove that for ( v , w ) ∈ V h × W h ( v , w ) ∈ V h × W h (v,w)inV_(h)xxW_(h)(v, w) \in V_{h} \times W_{h}(v,w)∈Vh×Wh,
A [ ( u h − Π h u , z h ) , ( v , w ) ] ≤ C h ‖ u ‖ ∗ ‖ ‖ ( v , w ) ‖ . A u h − Π h u , z h , ( v , w ) ≤ C h ‖ u ‖ ∗ ‖ ‖ ( v , w ) ‖ . A[(u_(h)-Pi_(h)u,z_(h)),(v,w)] <= Ch||u||_(**)||||(v,w)||.A\left[\left(u_{h}-\Pi_{h} u, z_{h}\right),(v, w)\right] \leq C h\|u\|_{*}\| \|(v, w) \| .A[(uh−Πhu,zh),(v,w)]≤Ch‖u‖∗‖‖(v,w)‖.
The weak form of (14) implies that
A [ ( u h − Π h u , z h ) , ( v , w ) ] = ( u − Π h u , v ) ω + G ( u − Π h u , w ) − s ( Π h u , v ) . A u h − Π h u , z h , ( v , w ) = u − Π h u , v ω + G u − Π h u , w − s Π h u , v . A[(u_(h)-Pi_(h)u,z_(h)),(v,w)]=(u-Pi_(h)u,v)_(omega)+G(u-Pi_(h)u,w)-s(Pi_(h)u,v).A\left[\left(u_{h}-\Pi_{h} u, z_{h}\right),(v, w)\right]=\left(u-\Pi_{h} u, v\right)_{\omega}+G\left(u-\Pi_{h} u, w\right)-s\left(\Pi_{h} u, v\right) .A[(uh−Πhu,zh),(v,w)]=(u−Πhu,v)ω+G(u−Πhu,w)−s(Πhu,v).
Using (16) we bound the first term to get
( u − Π h u , v ) ω ≤ C h 2 ‖ u ‖ H 2 ( Ω ) ‖ v ‖ ω u − Π h u , v ω ≤ C h 2 ‖ u ‖ H 2 ( Ω ) ‖ v ‖ ω (u-Pi_(h)u,v)_(omega) <= Ch^(2)||u||_(H^(2)(Omega))||v||_(omega)\left(u-\Pi_{h} u, v\right)_{\omega} \leq C h^{2}\|u\|_{H^{2}(\Omega)}\|v\|_{\omega}(u−Πhu,v)ω≤Ch2‖u‖H2(Ω)‖v‖ω
For the second term we use the L 2 L 2 L^(2)L^{2}L2-orthogonality property of Π h Π h Pi_(h)\Pi_{h}Πh, and (16) to obtain
G ( u − Π h u , w ) = ( ∇ ( u − Π h u ) , ∇ w ) L 2 ( Ω ) ≤ C h ‖ w ‖ W ‖ u ‖ H 2 ( Ω ) , G u − Π h u , w = ∇ u − Π h u , ∇ w L 2 ( Ω ) ≤ C h ‖ w ‖ W ‖ u ‖ H 2 ( Ω ) , G(u-Pi_(h)u,w)=(grad(u-Pi_(h)u),grad w)_(L^(2)(Omega)) <= Ch||w||_(W)||u||_(H^(2)(Omega)),G\left(u-\Pi_{h} u, w\right)=\left(\nabla\left(u-\Pi_{h} u\right), \nabla w\right)_{L^{2}(\Omega)} \leq C h\|w\|_{W}\|u\|_{H^{2}(\Omega)},G(u−Πhu,w)=(∇(u−Πhu),∇w)L2(Ω)≤Ch‖w‖W‖u‖H2(Ω),
while for the last term we employ (22) to estimate
s ( Π h u , v ) ≤ ‖ Π h u ‖ V ‖ v ‖ V ≤ C h ‖ u ‖ ∗ ‖ v ‖ V s Π h u , v ≤ Π h u V ‖ v ‖ V ≤ C h ‖ u ‖ ∗ ‖ v ‖ V s(Pi_(h)u,v) <= ||Pi_(h)u||_(V)||v||_(V) <= Ch||u||_(**)||v||_(V)s\left(\Pi_{h} u, v\right) \leq\left\|\Pi_{h} u\right\|_{V}\|v\|_{V} \leq C h\|u\|_{*}\|v\|_{V}s(Πhu,v)≤‖Πhu‖V‖v‖V≤Ch‖u‖∗‖v‖V ◻ ◻ ◻\square◻
Theorem 1. Let ω ⊂ B ⊂ Ω ω ⊂ B ⊂ Ω omega sub B sub Omega\omega \subset B \subset \Omegaω⊂B⊂Ω be defined as in Corollary 2. Let u ∈ H 2 ( Ω ) u ∈ H 2 ( Ω ) u inH^(2)(Omega)u \in H^{2}(\Omega)u∈H2(Ω) be the solution to (14) and ( u h , z h ) ∈ V h × W h u h , z h ∈ V h × W h (u_(h),z_(h))inV_(h)xxW_(h)\left(u_{h}, z_{h}\right) \in V_{h} \times W_{h}(uh,zh)∈Vh×Wh be the solution to (19). Then there are C > 0 C > 0 C > 0C>0C>0 and α ∈ ( 0 , 1 ) α ∈ ( 0 , 1 ) alpha in(0,1)\alpha \in(0,1)α∈(0,1) such that for all k , h > 0 k , h > 0 k,h > 0k, h>0k,h>0 with k h ≲ 1 k h ≲ 1 kh≲1k h \lesssim 1kh≲1
‖ u − u h ‖ L 2 ( B ) ≤ C ( h k ) α k α − 2 ‖ u ‖ ∗ . u − u h L 2 ( B ) ≤ C ( h k ) α k α − 2 ‖ u ‖ ∗ . ||u-u_(h)||_(L^(2)(B)) <= C(hk)^(alpha)k^(alpha-2)||u||_(**).\left\|u-u_{h}\right\|_{L^{2}(B)} \leq C(h k)^{\alpha} k^{\alpha-2}\|u\|_{*} .‖u−uh‖L2(B)≤C(hk)αkα−2‖u‖∗.
Proof. Consider the residual ⟨ r , w ⟩ = G ( u h − u , w ) = G ( u h , w ) − ⟨ f , w ⟩ , w ∈ H 0 1 ( Ω ) ⟨ r , w ⟩ = G u h − u , w = G u h , w − ⟨ f , w ⟩ , w ∈ H 0 1 ( Ω ) (:r,w:)=G(u_(h)-u,w)=G(u_(h),w)-(:f,w:),w inH_(0)^(1)(Omega)\langle r, w\rangle=G\left(u_{h}-u, w\right)=G\left(u_{h}, w\right)-\langle f, w\rangle, w \in H_{0}^{1}(\Omega)⟨r,w⟩=G(uh−u,w)=G(uh,w)−⟨f,w⟩,w∈H01(Ω). Taking v = 0 v = 0 v=0v=0v=0 in (19) we get G ( u h , w ) = ⟨ f , w ⟩ + s ∗ ( z h , w ) , w ∈ W h G u h , w = ⟨ f , w ⟩ + s ∗ z h , w , w ∈ W h G(u_(h),w)=(:f,w:)+s^(**)(z_(h),w),w inW_(h)G\left(u_{h}, w\right)=\langle f, w\rangle+s^{*}\left(z_{h}, w\right), w \in W_{h}G(uh,w)=⟨f,w⟩+s∗(zh,w),w∈Wh which implies that
⟨ r , w ⟩ = G ( u h , w ) − ⟨ f , w ⟩ − G ( u h , π h w ) + G ( u h , π h w ) = G ( u h , w − π h w ) − ⟨ f , w − π h w ⟩ + s ∗ ( z h , π h w ) , w ∈ H 0 1 ( Ω ) . ⟨ r , w ⟩ = G u h , w − ⟨ f , w ⟩ − G u h , π h w + G u h , π h w = G u h , w − π h w − f , w − π h w + s ∗ z h , π h w , w ∈ H 0 1 ( Ω ) . {:[(:r","w:)=G(u_(h),w)-(:f","w:)-G(u_(h),pi_(h)w)+G(u_(h),pi_(h)w)],[=G(u_(h),w-pi_(h)w)-(:f,w-pi_(h)w:)+s^(**)(z_(h),pi_(h)w)","quad w inH_(0)^(1)(Omega).]:}\begin{aligned} \langle r, w\rangle & =G\left(u_{h}, w\right)-\langle f, w\rangle-G\left(u_{h}, \pi_{h} w\right)+G\left(u_{h}, \pi_{h} w\right) \\ & =G\left(u_{h}, w-\pi_{h} w\right)-\left\langle f, w-\pi_{h} w\right\rangle+s^{*}\left(z_{h}, \pi_{h} w\right), \quad w \in H_{0}^{1}(\Omega) . \end{aligned}⟨r,w⟩=G(uh,w)−⟨f,w⟩−G(uh,πhw)+G(uh,πhw)=G(uh,w−πhw)−⟨f,w−πhw⟩+s∗(zh,πhw),w∈H01(Ω).
Using (21) and (16) we estimate the first term
G ( u h , w − π h w ) ≤ C ‖ u h ‖ V ( h − 1 ‖ w − π h w ‖ L 2 ( Ω ) + ‖ w − π h w ‖ H 1 ( Ω ) ) ≤ C ‖ u h ‖ V ‖ w ‖ H 1 ( Ω ) ≤ C h ‖ u ‖ ∗ ‖ w ‖ H 1 ( Ω ) G u h , w − π h w ≤ C u h V h − 1 w − π h w L 2 ( Ω ) + w − π h w H 1 ( Ω ) ≤ C u h V ‖ w ‖ H 1 ( Ω ) ≤ C h ‖ u ‖ ∗ ‖ w ‖ H 1 ( Ω ) {:[G(u_(h),w-pi_(h)w) <= C||u_(h)||_(V)(h^(-1)||w-pi_(h)w||_(L^(2)(Omega))+||w-pi_(h)w||_(H^(1)(Omega)))],[ <= C||u_(h)||_(V)||w||_(H^(1)(Omega)) <= Ch||u||_(**)||w||_(H^(1)(Omega))]:}\begin{aligned} G\left(u_{h}, w-\pi_{h} w\right) & \leq C\left\|u_{h}\right\|_{V}\left(h^{-1}\left\|w-\pi_{h} w\right\|_{L^{2}(\Omega)}+\left\|w-\pi_{h} w\right\|_{H^{1}(\Omega)}\right) \\ & \leq C\left\|u_{h}\right\|_{V}\|w\|_{H^{1}(\Omega)} \leq C h\|u\|_{*}\|w\|_{H^{1}(\Omega)} \end{aligned}G(uh,w−πhw)≤C‖uh‖V(h−1‖w−πhw‖L2(Ω)+‖w−πhw‖H1(Ω))≤C‖uh‖V‖w‖H1(Ω)≤Ch‖u‖∗‖w‖H1(Ω)
since, due to Lemma 4 and (22)
‖ u h ‖ V ≤ ‖ u h − Π h u ‖ V + ‖ Π h u ‖ V ≤ C h ‖ u ‖ ∗ . u h V ≤ u h − Π h u V + Π h u V ≤ C h ‖ u ‖ ∗ . ||u_(h)||_(V) <= ||u_(h)-Pi_(h)u||_(V)+||Pi_(h)u||_(V) <= Ch||u||_(**).\left\|u_{h}\right\|_{V} \leq\left\|u_{h}-\Pi_{h} u\right\|_{V}+\left\|\Pi_{h} u\right\|_{V} \leq C h\|u\|_{*} .‖uh‖V≤‖uh−Πhu‖V+‖Πhu‖V≤Ch‖u‖∗.
The second term is bounded by using (16)
⟨ f , w − π h w ⟩ ≤ ‖ f ‖ L 2 ( Ω ) ‖ w − π h w ‖ L 2 ( Ω ) ≤ C h ‖ f ‖ L 2 ( Ω ) ‖ w ‖ H 1 ( Ω ) f , w − π h w ≤ ‖ f ‖ L 2 ( Ω ) w − π h w L 2 ( Ω ) ≤ C h ‖ f ‖ L 2 ( Ω ) ‖ w ‖ H 1 ( Ω ) (:f,w-pi_(h)w:) <= ||f||_(L^(2)(Omega))||w-pi_(h)w||_(L^(2)(Omega)) <= Ch||f||_(L^(2)(Omega))||w||_(H^(1)(Omega))\left\langle f, w-\pi_{h} w\right\rangle \leq\|f\|_{L^{2}(\Omega)}\left\|w-\pi_{h} w\right\|_{L^{2}(\Omega)} \leq C h\|f\|_{L^{2}(\Omega)}\|w\|_{H^{1}(\Omega)}⟨f,w−πhw⟩≤‖f‖L2(Ω)‖w−πhw‖L2(Ω)≤Ch‖f‖L2(Ω)‖w‖H1(Ω)
and the last term by using Lemma 4 and the H 1 H 1 H^(1)H^{1}H1-stability (15)
s ∗ ( z h , π h w ) ≤ ‖ z h ‖ W ‖ π h w ‖ W ≤ C h ‖ u ‖ ∗ ‖ w ‖ H 1 ( Ω ) . s ∗ z h , π h w ≤ z h W π h w W ≤ C h ‖ u ‖ ∗ ‖ w ‖ H 1 ( Ω ) . s^(**)(z_(h),pi_(h)w) <= ||z_(h)||_(W)||pi_(h)w||_(W) <= Ch||u||_(**)||w||_(H^(1)(Omega)).s^{*}\left(z_{h}, \pi_{h} w\right) \leq\left\|z_{h}\right\|_{W}\left\|\pi_{h} w\right\|_{W} \leq C h\|u\|_{*}\|w\|_{H^{1}(\Omega)} .s∗(zh,πhw)≤‖zh‖W‖πhw‖W≤Ch‖u‖∗‖w‖H1(Ω).
Hence the following residual norm estimate holds
‖ r ‖ H − 1 ( Ω ) ≤ C h ( ‖ u ‖ ∗ + ‖ f ‖ L 2 ( Ω ) ) ≤ C h ‖ u ‖ ∗ . ‖ r ‖ H − 1 ( Ω ) ≤ C h ‖ u ‖ ∗ + ‖ f ‖ L 2 ( Ω ) ≤ C h ‖ u ‖ ∗ . ||r||_(H^(-1)(Omega)) <= Ch(||u||_(**)+||f||_(L^(2)(Omega))) <= Ch||u||_(**).\|r\|_{H^{-1}(\Omega)} \leq C h\left(\|u\|_{*}+\|f\|_{L^{2}(\Omega)}\right) \leq C h\|u\|_{*} .‖r‖H−1(Ω)≤Ch(‖u‖∗+‖f‖L2(Ω))≤Ch‖u‖∗.
Using the continuum estimate in Lemma 2 for u − u h u − u h u-u_(h)u-u_{h}u−uh we obtain the following error estimate
‖ u − u h ‖ L 2 ( B ) ≤ C ( ‖ u − u h ‖ L 2 ( ω ) + ‖ r ‖ H − 1 ( Ω ) ) α ‖ u − u h ‖ L 2 ( Ω ) 1 − α . u − u h L 2 ( B ) ≤ C u − u h L 2 ( ω ) + ‖ r ‖ H − 1 ( Ω ) α u − u h L 2 ( Ω ) 1 − α . ||u-u_(h)||_(L^(2)(B)) <= C(||u-u_(h)||_(L^(2)(omega))+||r||_(H^(-1)(Omega)))^(alpha)||u-u_(h)||_(L^(2)(Omega))^(1-alpha).\left\|u-u_{h}\right\|_{L^{2}(B)} \leq C\left(\left\|u-u_{h}\right\|_{L^{2}(\omega)}+\|r\|_{H^{-1}(\Omega)}\right)^{\alpha}\left\|u-u_{h}\right\|_{L^{2}(\Omega)}^{1-\alpha} .‖u−uh‖L2(B)≤C(‖u−uh‖L2(ω)+‖r‖H−1(Ω))α‖u−uh‖L2(Ω)1−α.
By (16) and Lemma 4 we have the bounds
‖ u − u h ‖ L 2 ( ω ) ≤ ‖ u − Π h u ‖ L 2 ( ω ) + ‖ u h − Π h u ‖ L 2 ( ω ) ≤ C h ‖ u ‖ H 1 ( Ω ) + C h ‖ u ‖ ∗ ≤ C h ‖ u ‖ ∗ u − u h L 2 ( ω ) ≤ u − Π h u L 2 ( ω ) + u h − Π h u L 2 ( ω ) ≤ C h ‖ u ‖ H 1 ( Ω ) + C h ‖ u ‖ ∗ ≤ C h ‖ u ‖ ∗ {:[||u-u_(h)||_(L^(2)(omega)) <= ||u-Pi_(h)u||_(L^(2)(omega))+||u_(h)-Pi_(h)u||_(L^(2)(omega))],[ <= Ch||u||_(H^(1)(Omega))+Ch||u||_(**)],[ <= Ch||u||_(**)]:}\begin{aligned} \left\|u-u_{h}\right\|_{L^{2}(\omega)} & \leq\left\|u-\Pi_{h} u\right\|_{L^{2}(\omega)}+\left\|u_{h}-\Pi_{h} u\right\|_{L^{2}(\omega)} \\ & \leq C h\|u\|_{H^{1}(\Omega)}+C h\|u\|_{*} \\ & \leq C h\|u\|_{*} \end{aligned}‖u−uh‖L2(ω)≤‖u−Πhu‖L2(ω)+‖uh−Πhu‖L2(ω)≤Ch‖u‖H1(Ω)+Ch‖u‖∗≤Ch‖u‖∗
and
‖ u − u h ‖ L 2 ( Ω ) ≤ ‖ u − Π h u ‖ L 2 ( Ω ) + ‖ u h − Π h u ‖ L 2 ( Ω ) ≤ C h 2 ‖ u ‖ H 2 ( Ω ) + C h − 1 k − 2 ‖ u h − Π h u ‖ V ≤ C ( ( h 2 + k − 2 ) ‖ u ‖ H 2 ( Ω ) + ‖ u ‖ L 2 ( Ω ) ) ≤ C k − 2 ‖ u ‖ ∗ u − u h L 2 ( Ω ) ≤ u − Π h u L 2 ( Ω ) + u h − Π h u L 2 ( Ω ) ≤ C h 2 ‖ u ‖ H 2 ( Ω ) + C h − 1 k − 2 u h − Π h u V ≤ C h 2 + k − 2 ‖ u ‖ H 2 ( Ω ) + ‖ u ‖ L 2 ( Ω ) ≤ C k − 2 ‖ u ‖ ∗ {:[||u-u_(h)||_(L^(2)(Omega)) <= ||u-Pi_(h)u||_(L^(2)(Omega))+||u_(h)-Pi_(h)u||_(L^(2)(Omega))],[ <= Ch^(2)||u||_(H^(2)(Omega))+Ch^(-1)k^(-2)||u_(h)-Pi_(h)u||_(V)],[ <= C((h^(2)+k^(-2))||u||_(H^(2)(Omega))+||u||_(L^(2)(Omega)))],[ <= Ck^(-2)||u||_(**)]:}\begin{aligned} \left\|u-u_{h}\right\|_{L^{2}(\Omega)} & \leq\left\|u-\Pi_{h} u\right\|_{L^{2}(\Omega)}+\left\|u_{h}-\Pi_{h} u\right\|_{L^{2}(\Omega)} \\ & \leq C h^{2}\|u\|_{H^{2}(\Omega)}+C h^{-1} k^{-2}\left\|u_{h}-\Pi_{h} u\right\|_{V} \\ & \leq C\left(\left(h^{2}+k^{-2}\right)\|u\|_{H^{2}(\Omega)}+\|u\|_{L^{2}(\Omega)}\right) \\ & \leq C k^{-2}\|u\|_{*} \end{aligned}‖u−uh‖L2(Ω)≤‖u−Πhu‖L2(Ω)+‖uh−Πhu‖L2(Ω)≤Ch2‖u‖H2(Ω)+Ch−1k−2‖uh−Πhu‖V≤C((h2+k−2)‖u‖H2(Ω)+‖u‖L2(Ω))≤Ck−2‖u‖∗
thus leading to the conclusion. ◻ ◻ ◻\square◻
Theorem 2. Let ω ⊂ B ⊂ Ω ω ⊂ B ⊂ Ω omega sub B sub Omega\omega \subset B \subset \Omegaω⊂B⊂Ω be defined as in Corollary 2. Let u ∈ H 2 ( Ω ) u ∈ H 2 ( Ω ) u inH^(2)(Omega)u \in H^{2}(\Omega)u∈H2(Ω) be the solution to (14) and ( u h , z h ) ∈ V h × W h u h , z h ∈ V h × W h (u_(h),z_(h))inV_(h)xxW_(h)\left(u_{h}, z_{h}\right) \in V_{h} \times W_{h}(uh,zh)∈Vh×Wh be the solution to (19). Then there are C > 0 C > 0 C > 0C>0C>0 and α ∈ ( 0 , 1 ) α ∈ ( 0 , 1 ) alpha in(0,1)\alpha \in(0,1)α∈(0,1) such that for all k , h > 0 k , h > 0 k,h > 0k, h>0k,h>0 with k h ≲ 1 k h ≲ 1 kh≲1k h \lesssim 1kh≲1
‖ u − u h ‖ H 1 ( B ) ≤ C ( h k ) α ‖ u ‖ ∗ u − u h H 1 ( B ) ≤ C ( h k ) α ‖ u ‖ ∗ ||u-u_(h)||_(H^(1)(B)) <= C(hk)^(alpha)||u||_(**)\left\|u-u_{h}\right\|_{H^{1}(B)} \leq C(h k)^{\alpha}\|u\|_{*}‖u−uh‖H1(B)≤C(hk)α‖u‖∗
Proof. We employ a similar argument as in the proof of Theorem 1 with the same estimates for the residual norm and the L 2 L 2 L^(2)L^{2}L2-errors in ω ω omega\omegaω and Ω Ω Omega\OmegaΩ, only now using the continuum estimate in Corollary 3 to obtain
‖ u − u h ‖ H 1 ( B ) ≤ C k ( ‖ u − u h ‖ L 2 ( ω ) + ‖ r ‖ H − 1 ( Ω ) ) α ( ‖ u − u h ‖ L 2 ( Ω ) + ‖ r ‖ H − 1 ( Ω ) ) 1 − α ≤ C k h α ( k − 2 + h ) 1 − α ‖ u ‖ ∗ u − u h H 1 ( B ) ≤ C k u − u h L 2 ( ω ) + ‖ r ‖ H − 1 ( Ω ) α u − u h L 2 ( Ω ) + ‖ r ‖ H − 1 ( Ω ) 1 − α ≤ C k h α k − 2 + h 1 − α ‖ u ‖ ∗ {:[||u-u_(h)||_(H^(1)(B)) <= Ck(||u-u_(h)||_(L^(2)(omega))+||r||_(H^(-1)(Omega)))^(alpha)(||u-u_(h)||_(L^(2)(Omega))+||r||_(H^(-1)(Omega)))^(1-alpha)],[ <= Ckh^(alpha)(k^(-2)+h)^(1-alpha)||u||_(**)]:}\begin{aligned} \left\|u-u_{h}\right\|_{H^{1}(B)} & \leq C k\left(\left\|u-u_{h}\right\|_{L^{2}(\omega)}+\|r\|_{H^{-1}(\Omega)}\right)^{\alpha}\left(\left\|u-u_{h}\right\|_{L^{2}(\Omega)}+\|r\|_{H^{-1}(\Omega)}\right)^{1-\alpha} \\ & \leq C k h^{\alpha}\left(k^{-2}+h\right)^{1-\alpha}\|u\|_{*} \end{aligned}‖u−uh‖H1(B)≤Ck(‖u−uh‖L2(ω)+‖r‖H−1(Ω))α(‖u−uh‖L2(Ω)+‖r‖H−1(Ω))1−α≤Ckhα(k−2+h)1−α‖u‖∗
which ends the proof.
Let us remark that if we make the assumption k 2 h ≲ 1 k 2 h ≲ 1 k^(2)h≲1k^{2} h \lesssim 1k2h≲1 then the estimate in Theorem 2 becomes
‖ u − u h ‖ H 1 ( B ) ≤ C ( h k 2 ) α k − 1 ‖ u ‖ ∗ u − u h H 1 ( B ) ≤ C h k 2 α k − 1 ‖ u ‖ ∗ ||u-u_(h)||_(H^(1)(B)) <= C(hk^(2))^(alpha)k^(-1)||u||_(**)\left\|u-u_{h}\right\|_{H^{1}(B)} \leq C\left(h k^{2}\right)^{\alpha} k^{-1}\|u\|_{*}‖u−uh‖H1(B)≤C(hk2)αk−1‖u‖∗
and combining Theorem 1 and Theorem 2 we obtain the following result.
Corollary 4. Let ω ⊂ B ⊂ Ω ω ⊂ B ⊂ Ω omega sub B sub Omega\omega \subset B \subset \Omegaω⊂B⊂Ω be defined as in Corollary 2. Let u ∈ H 2 ( Ω ) u ∈ H 2 ( Ω ) u inH^(2)(Omega)u \in H^{2}(\Omega)u∈H2(Ω) be the solution to (14) and ( u h , z h ) ∈ V h × W h u h , z h ∈ V h × W h (u_(h),z_(h))inV_(h)xxW_(h)\left(u_{h}, z_{h}\right) \in V_{h} \times W_{h}(uh,zh)∈Vh×Wh be the solution to (19). Then there are C > 0 C > 0 C > 0C>0C>0 and α ∈ ( 0 , 1 ) α ∈ ( 0 , 1 ) alpha in(0,1)\alpha \in(0,1)α∈(0,1) such that for all k , h > 0 k , h > 0 k,h > 0k, h>0k,h>0 with k 2 h ≲ 1 k 2 h ≲ 1 k^(2)h≲1k^{2} h \lesssim 1k2h≲1
k ‖ u − u h ‖ L 2 ( B ) + ‖ u − u h ‖ H 1 ( B ) ≤ C ( h k 2 ) α k − 1 ‖ u ‖ ∗ k u − u h L 2 ( B ) + u − u h H 1 ( B ) ≤ C h k 2 α k − 1 ‖ u ‖ ∗ k||u-u_(h)||_(L^(2)(B))+||u-u_(h)||_(H^(1)(B)) <= C(hk^(2))^(alpha)k^(-1)||u||_(**)k\left\|u-u_{h}\right\|_{L^{2}(B)}+\left\|u-u_{h}\right\|_{H^{1}(B)} \leq C\left(h k^{2}\right)^{\alpha} k^{-1}\|u\|_{*}k‖u−uh‖L2(B)+‖u−uh‖H1(B)≤C(hk2)αk−1‖u‖∗
Comparing with the well-posed boundary value problem (3) and the sharp bounds (4) and (5), we note that the k − 1 ‖ u ‖ ∗ k − 1 ‖ u ‖ ∗ k^(-1)||u||_(**)k^{-1}\|u\|_{*}k−1‖u‖∗ term in the above estimate is analogous to the well-posed case term ‖ f ‖ L 2 ( Ω ) ‖ f ‖ L 2 ( Ω ) ||f||_(L^(2)(Omega))\|f\|_{L^{2}(\Omega)}‖f‖L2(Ω).
3.3. Data perturbations. The analysis above can also handle the perturbed data
q ~ = q + δ q , f ~ = f + δ f q ~ = q + δ q , f ~ = f + δ f tilde(q)=q+delta q,quad tilde(f)=f+delta f\tilde{q}=q+\delta q, \quad \tilde{f}=f+\delta fq~=q+δq,f~=f+δf
with the unperturbed data q q qqq, f f fff in (14), and perturbations δ q ∈ L 2 ( ω ) , δ f ∈ H − 1 ( Ω ) δ q ∈ L 2 ( ω ) , δ f ∈ H − 1 ( Ω ) delta q inL^(2)(omega),delta f inH^(-1)(Omega)\delta q \in L^{2}(\omega), \delta f \in H^{-1}(\Omega)δq∈L2(ω),δf∈H−1(Ω) measured by
δ ( q ~ , f ~ ) = ‖ δ q ‖ ω + ‖ δ f ‖ H − 1 ( Ω ) δ ( q ~ , f ~ ) = ‖ δ q ‖ ω + ‖ δ f ‖ H − 1 ( Ω ) delta( tilde(q), tilde(f))=||delta q||_(omega)+||delta f||_(H^(-1)(Omega))\delta(\tilde{q}, \tilde{f})=\|\delta q\|_{\omega}+\|\delta f\|_{H^{-1}(\Omega)}δ(q~,f~)=‖δq‖ω+‖δf‖H−1(Ω)
The saddle points ( u , z ) ∈ V h × W h ( u , z ) ∈ V h × W h (u,z)inV_(h)xxW_(h)(u, z) \in V_{h} \times W_{h}(u,z)∈Vh×Wh of the perturbed Lagrangian L q ~ , f ~ L q ~ , f ~ L_( tilde(q), tilde(f))L_{\tilde{q}, \tilde{f}}Lq~,f~ satisfy
(23) A [ ( u , z ) , ( v , w ) ] = ( q ~ , v ) ω + ⟨ f ~ , w ⟩ , ( v , w ) ∈ V h × W h (23) A [ ( u , z ) , ( v , w ) ] = ( q ~ , v ) ω + ⟨ f ~ , w ⟩ , ( v , w ) ∈ V h × W h {:(23)A[(u","z)","(v","w)]=( tilde(q)","v)_(omega)+(: tilde(f)","w:)","quad(v","w)inV_(h)xxW_(h):}\begin{equation*} A[(u, z),(v, w)]=(\tilde{q}, v)_{\omega}+\langle\tilde{f}, w\rangle, \quad(v, w) \in V_{h} \times W_{h} \tag{23} \end{equation*}(23)A[(u,z),(v,w)]=(q~,v)ω+⟨f~,w⟩,(v,w)∈Vh×Wh
Lemma 5. Let u ∈ H 2 ( Ω ) u ∈ H 2 ( Ω ) u inH^(2)(Omega)u \in H^{2}(\Omega)u∈H2(Ω) be the solution to the unperturbed problem (14) and ( u h , z h ) ∈ V h × W h u h , z h ∈ V h × W h (u_(h),z_(h))inV_(h)xxW_(h)\left(u_{h}, z_{h}\right) \in V_{h} \times W_{h}(uh,zh)∈Vh×Wh be the solution to the perturbed problem (23). Then there exists C > 0 C > 0 C > 0C>0C>0 such that for all h ∈ ( 0 , 1 ) h ∈ ( 0 , 1 ) h in(0,1)h \in(0,1)h∈(0,1)
‖ ∣ ( u h − Π h u , z h ) ‖ ≤ C ( h ‖ u ‖ ∗ + δ ( q ~ , f ~ ) ) ∣ u h − Π h u , z h ≤ C h ‖ u ‖ ∗ + δ ( q ~ , f ~ ) ||∣(u_(h)-Pi_(h)u,z_(h))|| <= C(h||u||_(**)+delta(( tilde(q)),( tilde(f))))\left\|\mid\left(u_{h}-\Pi_{h} u, z_{h}\right)\right\| \leq C\left(h\|u\|_{*}+\delta(\tilde{q}, \tilde{f})\right)‖∣(uh−Πhu,zh)‖≤C(h‖u‖∗+δ(q~,f~))
Proof. Proceeding as in the proof of Lemma 4, the weak form gives
A [ ( u h − Π h u , z h ) , ( v , w ) ] = ( u − Π h u , v ) ω + G ( u − Π h u , w ) − s ( Π h u , v ) + ( δ q , v ) ω + ⟨ δ f , w ⟩ A u h − Π h u , z h , ( v , w ) = u − Π h u , v ω + G u − Π h u , w − s Π h u , v + ( δ q , v ) ω + ⟨ δ f , w ⟩ {:[A[(u_(h)-Pi_(h)u,z_(h)),(v,w)]=(u-Pi_(h)u,v)_(omega)+G(u-Pi_(h)u,w)-s(Pi_(h)u,v)],[+(delta q","v)_(omega)+(:delta f","w:)]:}\begin{aligned} A\left[\left(u_{h}-\Pi_{h} u, z_{h}\right),(v, w)\right] & =\left(u-\Pi_{h} u, v\right)_{\omega}+G\left(u-\Pi_{h} u, w\right)-s\left(\Pi_{h} u, v\right) \\ & +(\delta q, v)_{\omega}+\langle\delta f, w\rangle \end{aligned}A[(uh−Πhu,zh),(v,w)]=(u−Πhu,v)ω+G(u−Πhu,w)−s(Πhu,v)+(δq,v)ω+⟨δf,w⟩
We bound the perturbation terms by
( δ q , v ) ω + ⟨ δ f , w ⟩ ≤ ‖ δ q ‖ ω ‖ v ‖ ω + C ‖ δ f ‖ H − 1 ( Ω ) ‖ w ‖ W ≤ C δ ( q ~ , f ~ ) ‖ ( v , w ) ‖ ( δ q , v ) ω + ⟨ δ f , w ⟩ ≤ ‖ δ q ‖ ω ‖ v ‖ ω + C ‖ δ f ‖ H − 1 ( Ω ) ‖ w ‖ W ≤ C δ ( q ~ , f ~ ) ‖ ( v , w ) ‖ {:[(delta q","v)_(omega)+(:delta f","w:) <= ||delta q||_(omega)||v||_(omega)+C||delta f||_(H^(-1)(Omega))||w||_(W)],[ <= C delta( tilde(q)"," tilde(f))||(v","w)||]:}\begin{aligned} (\delta q, v)_{\omega}+\langle\delta f, w\rangle & \leq\|\delta q\|_{\omega}\|v\|_{\omega}+C\|\delta f\|_{H^{-1}(\Omega)}\|w\|_{W} \\ & \leq C \delta(\tilde{q}, \tilde{f})\|(v, w)\| \end{aligned}(δq,v)ω+⟨δf,w⟩≤‖δq‖ω‖v‖ω+C‖δf‖H−1(Ω)‖w‖W≤Cδ(q~,f~)‖(v,w)‖
and we conclude by using the previously derived bounds for the other terms.
Theorem 3. Let ω ⊂ B ⊂ Ω ω ⊂ B ⊂ Ω omega sub B sub Omega\omega \subset B \subset \Omegaω⊂B⊂Ω be defined as in Corollary 2. Let u ∈ H 2 ( Ω ) u ∈ H 2 ( Ω ) u inH^(2)(Omega)u \in H^{2}(\Omega)u∈H2(Ω) be the solution to the unperturbed problem (14) and ( u h , z h ) ∈ V h × W h u h , z h ∈ V h × W h (u_(h),z_(h))inV_(h)xxW_(h)\left(u_{h}, z_{h}\right) \in V_{h} \times W_{h}(uh,zh)∈Vh×Wh be the solution to the perturbed problem (23). Then there are C > 0 C > 0 C > 0C>0C>0 and α ∈ ( 0 , 1 ) α ∈ ( 0 , 1 ) alpha in(0,1)\alpha \in(0,1)α∈(0,1) such that for all k , h > 0 k , h > 0 k,h > 0k, h>0k,h>0 with k h ≲ 1 k h ≲ 1 kh≲1k h \lesssim 1kh≲1
‖ u − u h ‖ L 2 ( B ) ≤ C ( h k ) α k α − 2 ( ‖ u ‖ ∗ + h − 1 δ ( q ~ , f ~ ) ) u − u h L 2 ( B ) ≤ C ( h k ) α k α − 2 ‖ u ‖ ∗ + h − 1 δ ( q ~ , f ~ ) ||u-u_(h)||_(L^(2)(B)) <= C(hk)^(alpha)k^(alpha-2)(||u||_(**)+h^(-1)delta(( tilde(q)),( tilde(f))))\left\|u-u_{h}\right\|_{L^{2}(B)} \leq C(h k)^{\alpha} k^{\alpha-2}\left(\|u\|_{*}+h^{-1} \delta(\tilde{q}, \tilde{f})\right)‖u−uh‖L2(B)≤C(hk)αkα−2(‖u‖∗+h−1δ(q~,f~))
Proof. Following the proof of Theorem 1, the residual satisfies
⟨ r , w ⟩ = G ( u h , w − π h w ) − ⟨ f , w − π h w ⟩ + s ∗ ( z h , π h w ) + ⟨ δ f , π h w ⟩ , w ∈ H 0 1 ( Ω ) ⟨ r , w ⟩ = G u h , w − π h w − f , w − π h w + s ∗ z h , π h w + δ f , π h w , w ∈ H 0 1 ( Ω ) (:r,w:)=G(u_(h),w-pi_(h)w)-(:f,w-pi_(h)w:)+s^(**)(z_(h),pi_(h)w)+(:delta f,pi_(h)w:),quad w inH_(0)^(1)(Omega)\langle r, w\rangle=G\left(u_{h}, w-\pi_{h} w\right)-\left\langle f, w-\pi_{h} w\right\rangle+s^{*}\left(z_{h}, \pi_{h} w\right)+\left\langle\delta f, \pi_{h} w\right\rangle, \quad w \in H_{0}^{1}(\Omega)⟨r,w⟩=G(uh,w−πhw)−⟨f,w−πhw⟩+s∗(zh,πhw)+⟨δf,πhw⟩,w∈H01(Ω)
and
‖ r ‖ H − 1 ( Ω ) ≤ C ( ‖ u h ‖ V + h ‖ f ‖ L 2 ( Ω ) + ‖ z h ‖ W + ‖ δ f ‖ H − 1 ( Ω ) ) . ‖ r ‖ H − 1 ( Ω ) ≤ C u h V + h ‖ f ‖ L 2 ( Ω ) + z h W + ‖ δ f ‖ H − 1 ( Ω ) . ||r||_(H^(-1)(Omega)) <= C(||u_(h)||_(V)+h||f||_(L^(2)(Omega))+||z_(h)||_(W)+||delta f||_(H^(-1)(Omega))).\|r\|_{H^{-1}(\Omega)} \leq C\left(\left\|u_{h}\right\|_{V}+h\|f\|_{L^{2}(\Omega)}+\left\|z_{h}\right\|_{W}+\|\delta f\|_{H^{-1}(\Omega)}\right) .‖r‖H−1(Ω)≤C(‖uh‖V+h‖f‖L2(Ω)+‖zh‖W+‖δf‖H−1(Ω)).
Bounding the first term in the right-hand side by Lemma 5 and (22)
‖ u h ‖ V ≤ ‖ u h − Π h u ‖ V + ‖ Π h u ‖ V ≤ C ( h ‖ u ‖ ∗ + δ ( q ~ , f ~ ) ) u h V ≤ u h − Π h u V + Π h u V ≤ C h ‖ u ‖ ∗ + δ ( q ~ , f ~ ) ||u_(h)||_(V) <= ||u_(h)-Pi_(h)u||_(V)+||Pi_(h)u||_(V) <= C(h||u||_(**)+delta(( tilde(q)),( tilde(f))))\left\|u_{h}\right\|_{V} \leq\left\|u_{h}-\Pi_{h} u\right\|_{V}+\left\|\Pi_{h} u\right\|_{V} \leq C\left(h\|u\|_{*}+\delta(\tilde{q}, \tilde{f})\right)‖uh‖V≤‖uh−Πhu‖V+‖Πhu‖V≤C(h‖u‖∗+δ(q~,f~))
and the third one by Lemma 5 again, we obtain
‖ r ‖ H − 1 ( Ω ) ≤ C h ( ‖ u ‖ ∗ + ‖ f ‖ L 2 ( Ω ) ) + C δ ( q ~ , f ~ ) ≤ C ( h ‖ u ‖ ∗ + δ ( q ~ , f ~ ) ) ‖ r ‖ H − 1 ( Ω ) ≤ C h ‖ u ‖ ∗ + ‖ f ‖ L 2 ( Ω ) + C δ ( q ~ , f ~ ) ≤ C h ‖ u ‖ ∗ + δ ( q ~ , f ~ ) ||r||_(H^(-1)(Omega)) <= Ch(||u||_(**)+||f||_(L^(2)(Omega)))+C delta( tilde(q), tilde(f)) <= C(h||u||_(**)+delta(( tilde(q)),( tilde(f))))\|r\|_{H^{-1}(\Omega)} \leq C h\left(\|u\|_{*}+\|f\|_{L^{2}(\Omega)}\right)+C \delta(\tilde{q}, \tilde{f}) \leq C\left(h\|u\|_{*}+\delta(\tilde{q}, \tilde{f})\right)‖r‖H−1(Ω)≤Ch(‖u‖∗+‖f‖L2(Ω))+Cδ(q~,f~)≤C(h‖u‖∗+δ(q~,f~))
The continuum estimate in Lemma 2 applied to u − u h u − u h u-u_(h)u-u_{h}u−uh gives
‖ u − u h ‖ L 2 ( B ) ≤ C ( h ‖ u ‖ ∗ + δ ( q ~ , f ~ ) ) α ‖ u − u h ‖ L 2 ( Ω ) 1 − α u − u h L 2 ( B ) ≤ C h ‖ u ‖ ∗ + δ ( q ~ , f ~ ) α u − u h L 2 ( Ω ) 1 − α ||u-u_(h)||_(L^(2)(B)) <= C(h||u||_(**)+delta(( tilde(q)),( tilde(f))))^(alpha)||u-u_(h)||_(L^(2)(Omega))^(1-alpha)\left\|u-u_{h}\right\|_{L^{2}(B)} \leq C\left(h\|u\|_{*}+\delta(\tilde{q}, \tilde{f})\right)^{\alpha}\left\|u-u_{h}\right\|_{L^{2}(\Omega)}^{1-\alpha}‖u−uh‖L2(B)≤C(h‖u‖∗+δ(q~,f~))α‖u−uh‖L2(Ω)1−α
where ‖ u − u h ‖ L 2 ( ω ) u − u h L 2 ( ω ) ||u-u_(h)||_(L^(2)(omega))\left\|u-u_{h}\right\|_{L^{2}(\omega)}‖u−uh‖L2(ω) was bounded by using Lemma 5 and (16). Then the bound
‖ u − u h ‖ L 2 ( Ω ) ≤ ‖ u − Π h u ‖ L 2 ( Ω ) + ‖ u h − Π h u ‖ L 2 ( Ω ) ≤ C ( h 2 ‖ u ‖ H 2 ( Ω ) + h − 1 k − 2 ‖ u h − Π h u ‖ V ) ≤ C ( h 2 ‖ u ‖ H 2 ( Ω ) + k − 2 ‖ u ‖ ∗ + h − 1 k − 2 δ ( q ~ , f ~ ) ) ≤ C k − 2 ( ‖ u ‖ ∗ + h − 1 δ ( q ~ , f ~ ) ) u − u h L 2 ( Ω ) ≤ u − Π h u L 2 ( Ω ) + u h − Π h u L 2 ( Ω ) ≤ C h 2 ‖ u ‖ H 2 ( Ω ) + h − 1 k − 2 u h − Π h u V ≤ C h 2 ‖ u ‖ H 2 ( Ω ) + k − 2 ‖ u ‖ ∗ + h − 1 k − 2 δ ( q ~ , f ~ ) ≤ C k − 2 ‖ u ‖ ∗ + h − 1 δ ( q ~ , f ~ ) {:[||u-u_(h)||_(L^(2)(Omega)) <= ||u-Pi_(h)u||_(L^(2)(Omega))+||u_(h)-Pi_(h)u||_(L^(2)(Omega))],[ <= C(h^(2)||u||_(H^(2)(Omega))+h^(-1)k^(-2)||u_(h)-Pi_(h)u||_(V))],[ <= C(h^(2)||u||_(H^(2)(Omega))+k^(-2)||u||_(**)+h^(-1)k^(-2)delta(( tilde(q)),( tilde(f))))],[ <= Ck^(-2)(||u||_(**)+h^(-1)delta(( tilde(q)),( tilde(f))))]:}\begin{aligned} \left\|u-u_{h}\right\|_{L^{2}(\Omega)} & \leq\left\|u-\Pi_{h} u\right\|_{L^{2}(\Omega)}+\left\|u_{h}-\Pi_{h} u\right\|_{L^{2}(\Omega)} \\ & \leq C\left(h^{2}\|u\|_{H^{2}(\Omega)}+h^{-1} k^{-2}\left\|u_{h}-\Pi_{h} u\right\|_{V}\right) \\ & \leq C\left(h^{2}\|u\|_{H^{2}(\Omega)}+k^{-2}\|u\|_{*}+h^{-1} k^{-2} \delta(\tilde{q}, \tilde{f})\right) \\ & \leq C k^{-2}\left(\|u\|_{*}+h^{-1} \delta(\tilde{q}, \tilde{f})\right) \end{aligned}‖u−uh‖L2(Ω)≤‖u−Πhu‖L2(Ω)+‖uh−Πhu‖L2(Ω)≤C(h2‖u‖H2(Ω)+h−1k−2‖uh−Πhu‖V)≤C(h2‖u‖H2(Ω)+k−2‖u‖∗+h−1k−2δ(q~,f~))≤Ck−2(‖u‖∗+h−1δ(q~,f~))
concludes the proof.
Theorem 4. Let ω ⊂ B ⊂ Ω ω ⊂ B ⊂ Ω omega sub B sub Omega\omega \subset B \subset \Omegaω⊂B⊂Ω be defined as in Corollary 2. Let u ∈ H 2 ( Ω ) u ∈ H 2 ( Ω ) u inH^(2)(Omega)u \in H^{2}(\Omega)u∈H2(Ω) be the solution to the unperturbed problem (14) and ( u h , z h ) ∈ V h × W h u h , z h ∈ V h × W h (u_(h),z_(h))inV_(h)xxW_(h)\left(u_{h}, z_{h}\right) \in V_{h} \times W_{h}(uh,zh)∈Vh×Wh be the solution to the perturbed problem (23). Then there are C > 0 C > 0 C > 0C>0C>0 and α ∈ ( 0 , 1 ) α ∈ ( 0 , 1 ) alpha in(0,1)\alpha \in(0,1)α∈(0,1) such that for all k , h > 0 k , h > 0 k,h > 0k, h>0k,h>0 with k h ≲ 1 k h ≲ 1 kh≲1k h \lesssim 1kh≲1
‖ u − u h ‖ H 1 ( B ) ≤ C ( h k ) α ( ‖ u ‖ ∗ + h − 1 δ ( q ~ , f ~ ) ) u − u h H 1 ( B ) ≤ C ( h k ) α ‖ u ‖ ∗ + h − 1 δ ( q ~ , f ~ ) ||u-u_(h)||_(H^(1)(B)) <= C(hk)^(alpha)(||u||_(**)+h^(-1)delta(( tilde(q)),( tilde(f))))\left\|u-u_{h}\right\|_{H^{1}(B)} \leq C(h k)^{\alpha}\left(\|u\|_{*}+h^{-1} \delta(\tilde{q}, \tilde{f})\right)‖u−uh‖H1(B)≤C(hk)α(‖u‖∗+h−1δ(q~,f~))
Proof. Following the proof of Theorem 3, we now use Corollary 3 to derive
‖ u − u h ‖ H 1 ( B ) ≤ C k ( ‖ u − u h ‖ L 2 ( ω ) + ‖ r ‖ H − 1 ( Ω ) ) α ( ‖ u − u h ‖ L 2 ( Ω ) + ‖ r ‖ H − 1 ( Ω ) ) 1 − α ≤ C k ( h ‖ u ‖ ∗ + δ ( q ~ , f ~ ) ) α ( ( k − 2 + h ) ( ‖ u ‖ ∗ + h − 1 δ ( q ~ , f ~ ) ) ) 1 − α ≤ C k h α ( k − 2 + h ) 1 − α ( ‖ u ‖ ∗ + h − 1 δ ( q ~ , f ~ ) ) u − u h H 1 ( B ) ≤ C k u − u h L 2 ( ω ) + ‖ r ‖ H − 1 ( Ω ) α u − u h L 2 ( Ω ) + ‖ r ‖ H − 1 ( Ω ) 1 − α ≤ C k h ‖ u ‖ ∗ + δ ( q ~ , f ~ ) α k − 2 + h ‖ u ‖ ∗ + h − 1 δ ( q ~ , f ~ ) 1 − α ≤ C k h α k − 2 + h 1 − α ‖ u ‖ ∗ + h − 1 δ ( q ~ , f ~ ) {:[||u-u_(h)||_(H^(1)(B)) <= Ck(||u-u_(h)||_(L^(2)(omega))+||r||_(H^(-1)(Omega)))^(alpha)(||u-u_(h)||_(L^(2)(Omega))+||r||_(H^(-1)(Omega)))^(1-alpha)],[ <= Ck(h||u||_(**)+delta(( tilde(q)),( tilde(f))))^(alpha)((k^(-2)+h)(||u||_(**)+h^(-1)delta(( tilde(q)),( tilde(f)))))^(1-alpha)],[ <= Ckh^(alpha)(k^(-2)+h)^(1-alpha)(||u||_(**)+h^(-1)delta(( tilde(q)),( tilde(f))))]:}\begin{aligned} \left\|u-u_{h}\right\|_{H^{1}(B)} & \leq C k\left(\left\|u-u_{h}\right\|_{L^{2}(\omega)}+\|r\|_{H^{-1}(\Omega)}\right)^{\alpha}\left(\left\|u-u_{h}\right\|_{L^{2}(\Omega)}+\|r\|_{H^{-1}(\Omega)}\right)^{1-\alpha} \\ & \leq C k\left(h\|u\|_{*}+\delta(\tilde{q}, \tilde{f})\right)^{\alpha}\left(\left(k^{-2}+h\right)\left(\|u\|_{*}+h^{-1} \delta(\tilde{q}, \tilde{f})\right)\right)^{1-\alpha} \\ & \leq C k h^{\alpha}\left(k^{-2}+h\right)^{1-\alpha}\left(\|u\|_{*}+h^{-1} \delta(\tilde{q}, \tilde{f})\right) \end{aligned}‖u−uh‖H1(B)≤Ck(‖u−uh‖L2(ω)+‖r‖H−1(Ω))α(‖u−uh‖L2(Ω)+‖r‖H−1(Ω))1−α≤Ck(h‖u‖∗+δ(q~,f~))α((k−2+h)(‖u‖∗+h−1δ(q~,f~)))1−α≤Ckhα(k−2+h)1−α(‖u‖∗+h−1δ(q~,f~))
which ends the proof.
Analogous to the unpolluted case, if k 2 h ≲ 1 k 2 h ≲ 1 k^(2)h≲1k^{2} h \lesssim 1k2h≲1 the above result becomes
‖ u − u h ‖ H 1 ( B ) ≤ C ( h k 2 ) α k − 1 ( ‖ u ‖ ∗ + h − 1 δ ( q ~ , f ~ ) ) u − u h H 1 ( B ) ≤ C h k 2 α k − 1 ‖ u ‖ ∗ + h − 1 δ ( q ~ , f ~ ) ||u-u_(h)||_(H^(1)(B)) <= C(hk^(2))^(alpha)k^(-1)(||u||_(**)+h^(-1)delta(( tilde(q)),( tilde(f))))\left\|u-u_{h}\right\|_{H^{1}(B)} \leq C\left(h k^{2}\right)^{\alpha} k^{-1}\left(\|u\|_{*}+h^{-1} \delta(\tilde{q}, \tilde{f})\right)‖u−uh‖H1(B)≤C(hk2)αk−1(‖u‖∗+h−1δ(q~,f~))
and combining Theorem 3 and Theorem 4 gives the following.
Corollary 5. Let ω ⊂ B ⊂ Ω ω ⊂ B ⊂ Ω omega sub B sub Omega\omega \subset B \subset \Omegaω⊂B⊂Ω be defined as in Corollary 2. Let u ∈ H 2 ( Ω ) u ∈ H 2 ( Ω ) u inH^(2)(Omega)u \in H^{2}(\Omega)u∈H2(Ω) be the solution to the unperturbed problem (14) and ( u h , z h ) ∈ V h × W h u h , z h ∈ V h × W h (u_(h),z_(h))inV_(h)xxW_(h)\left(u_{h}, z_{h}\right) \in V_{h} \times W_{h}(uh,zh)∈Vh×Wh be the solution to the perturbed problem (23). Then there are C > 0 C > 0 C > 0C>0C>0 and α ∈ ( 0 , 1 ) α ∈ ( 0 , 1 ) alpha in(0,1)\alpha \in(0,1)α∈(0,1) such that for all k , h > 0 k , h > 0 k,h > 0k, h>0k,h>0 with k 2 h ≲ 1 k 2 h ≲ 1 k^(2)h≲1k^{2} h \lesssim 1k2h≲1
k ‖ u − u h ‖ L 2 ( B ) + ‖ u − u h ‖ H 1 ( B ) ≤ C ( h k 2 ) α k − 1 ( ‖ u ‖ ∗ + h − 1 δ ( q ~ , f ~ ) ) . k u − u h L 2 ( B ) + u − u h H 1 ( B ) ≤ C h k 2 α k − 1 ‖ u ‖ ∗ + h − 1 δ ( q ~ , f ~ ) . k||u-u_(h)||_(L^(2)(B))+||u-u_(h)||_(H^(1)(B)) <= C(hk^(2))^(alpha)k^(-1)(||u||_(**)+h^(-1)delta(( tilde(q)),( tilde(f)))).k\left\|u-u_{h}\right\|_{L^{2}(B)}+\left\|u-u_{h}\right\|_{H^{1}(B)} \leq C\left(h k^{2}\right)^{\alpha} k^{-1}\left(\|u\|_{*}+h^{-1} \delta(\tilde{q}, \tilde{f})\right) .k‖u−uh‖L2(B)+‖u−uh‖H1(B)≤C(hk2)αk−1(‖u‖∗+h−1δ(q~,f~)).

4. Numerical examples

We illustrate the above theoretical results for the unique continuation problem (14) with some numerical examples. Drawing on previous results in [5], we adjust the stabilizer in (19) with a fixed stabilization parameter γ > 0 γ > 0 gamma > 0\gamma>0γ>0 such that s ( u , v ) = γ J ( u , v ) + γ h 2 k 4 ( u , v ) L 2 ( Ω ) s ( u , v ) = γ J ( u , v ) + γ h 2 k 4 ( u , v ) L 2 ( Ω ) s(u,v)=gammaJ(u,v)+gammah^(2)k^(4)(u,v)_(L^(2)(Omega))s(u, v)=\gamma \mathcal{J}(u, v)+\gamma h^{2} k^{4}(u, v)_{L^{2}(\Omega)}s(u,v)=γJ(u,v)+γh2k4(u,v)L2(Ω). The error analysis stays unchanged under this rescaling. Various numerical experiments indicate that γ = 10 − 5 γ = 10 − 5 gamma=10^(-5)\gamma=10^{-5}γ=10−5 is a near-optimal value for different kinds of geometries and solutions. The implementation of our method and all the computations have been carried out in FreeFem++ [14]. The domain Ω Ω Omega\OmegaΩ is the unit square, and the triangulation is uniform with alternating left and right diagonals, as shown in Figure 2. The mesh size is taken as the inverse square root of the number of nodes.
In the light of the convexity assumptions in Section 2, we shall consider two different geometric settings: one in which the data is continued in the convex direction, inside the convex hull of ω ω omega\omegaω, and one in which the solution is continued in the non-convex direction, outside the convex hull of ω ω omega\omegaω.
In the convex setting, given in Figure 3a, we take
(24) ω = Ω ∖ [ 0.1 , 0.9 ] × [ 0.25 , 1 ] , B = Ω ∖ [ 0.1 , 0.9 ] × [ 0.95 , 1 ] (24) ω = Ω ∖ [ 0.1 , 0.9 ] × [ 0.25 , 1 ] , B = Ω ∖ [ 0.1 , 0.9 ] × [ 0.95 , 1 ] {:(24)omega=Omega\\[0.1","0.9]xx[0.25","1]","quad B=Omega\\[0.1","0.9]xx[0.95","1]:}\begin{equation*} \omega=\Omega \backslash[0.1,0.9] \times[0.25,1], \quad B=\Omega \backslash[0.1,0.9] \times[0.95,1] \tag{24} \end{equation*}(24)ω=Ω∖[0.1,0.9]×[0.25,1],B=Ω∖[0.1,0.9]×[0.95,1]
for continuing the solution inside the convex hull of ω ω omega\omegaω. This example does not correspond exactly to the specific geometric setting in Corollary 2, but all the theoretical results are valid in this case as proven in Example 1. below.
Example 1. Let ω ⊂ B ⊂ Ω ω ⊂ B ⊂ Ω omega sub B sub Omega\omega \subset B \subset \Omegaω⊂B⊂Ω be defined by (24) (Figure 3a). Then the stability estimates in Corollary 2, Corollary 3 and Lemma 2 hold true.
Proof. Consider an extended rectangle Ω ~ ⊃ Ω Ω ~ ⊃ Ω tilde(Omega)sup Omega\tilde{\Omega} \supset \OmegaΩ~⊃Ω such that the unit square Ω Ω Omega\OmegaΩ is centred horizontally and touches the upper side of Ω ~ Ω ~ tilde(Omega)\tilde{\Omega}Ω~, and ω ~ ⊃ ω ω ~ ⊃ ω tilde(omega)sup omega\tilde{\omega} \supset \omegaω~⊃ω and B ~ ⊃ B B ~ ⊃ B tilde(B)sup B\tilde{B} \supset BB~⊃B are defined as in Corollary 2. Choose a smooth cutoff function χ χ chi\chiχ such that χ = 1 χ = 1 chi=1\chi=1χ=1 in Ω ∖ ω Ω ∖ ω Omega\\omega\Omega \backslash \omegaΩ∖ω and χ = 0 χ = 0 chi=0\chi=0χ=0 in Ω ~ ∖ Ω Ω ~ ∖ Ω tilde(Omega)\\Omega\tilde{\Omega} \backslash \OmegaΩ~∖Ω. Applying now Corollary 2 for ω ~ , B ~ , Ω ~ ω ~ , B ~ , Ω ~ tilde(omega), tilde(B), tilde(Omega)\tilde{\omega}, \tilde{B}, \tilde{\Omega}ω~,B~,Ω~ and χ u χ u chi u\chi uχu we get
‖ u ‖ H 1 ( B ∖ ω ) ≤ C ‖ χ u ‖ H 1 ( B ~ ∖ ω ~ ) ≤ C ( ‖ χ u ‖ H 1 ( ω ~ ) + ‖ Δ ( χ u ) + k 2 χ u ‖ L 2 ( Ω ~ ) ) α ‖ χ u ‖ H 1 ( Ω ~ ) 1 − α ≤ C ( ‖ u ‖ H 1 ( ω ) + ‖ Δ u + k 2 u ‖ L 2 ( Ω ) ) α ‖ u ‖ H 1 ( Ω ) 1 − α ‖ u ‖ H 1 ( B ∖ ω ) ≤ C ‖ χ u ‖ H 1 ( B ~ ∖ ω ~ ) ≤ C ‖ χ u ‖ H 1 ( ω ~ ) + Δ ( χ u ) + k 2 χ u L 2 ( Ω ~ ) α ‖ χ u ‖ H 1 ( Ω ~ ) 1 − α ≤ C ‖ u ‖ H 1 ( ω ) + Δ u + k 2 u L 2 ( Ω ) α ‖ u ‖ H 1 ( Ω ) 1 − α {:[||u||_(H^(1)(B\\omega)) <= C||chi u||_(H^(1)( tilde(B)\\ tilde(omega))) <= C(||chi u||_(H^(1)( tilde(omega)))+||Delta(chi u)+k^(2)chi u||_(L^(2)( tilde(Omega))))^(alpha)||chi u||_(H^(1)( tilde(Omega)))^(1-alpha)],[ <= C(||u||_(H^(1)(omega))+||Delta u+k^(2)u||_(L^(2)(Omega)))^(alpha)||u||_(H^(1)(Omega))^(1-alpha)]:}\begin{aligned} \|u\|_{H^{1}(B \backslash \omega)} & \leq C\|\chi u\|_{H^{1}(\tilde{B} \backslash \tilde{\omega})} \leq C\left(\|\chi u\|_{H^{1}(\tilde{\omega})}+\left\|\Delta(\chi u)+k^{2} \chi u\right\|_{L^{2}(\tilde{\Omega})}\right)^{\alpha}\|\chi u\|_{H^{1}(\tilde{\Omega})}^{1-\alpha} \\ & \leq C\left(\|u\|_{H^{1}(\omega)}+\left\|\Delta u+k^{2} u\right\|_{L^{2}(\Omega)}\right)^{\alpha}\|u\|_{H^{1}(\Omega)}^{1-\alpha} \end{aligned}‖u‖H1(B∖ω)≤C‖χu‖H1(B~∖ω~)≤C(‖χu‖H1(ω~)+‖Δ(χu)+k2χu‖L2(Ω~))α‖χu‖H1(Ω~)1−α≤C(‖u‖H1(ω)+‖Δu+k2u‖L2(Ω))α‖u‖H1(Ω)1−α
where we have used that the commutator [ Δ , χ ] u [ Δ , χ ] u [Delta,chi]u[\Delta, \chi] u[Δ,χ]u is supported in ω ω omega\omegaω. A similar proof is valid for the estimates in Corollary 3 and Lemma 2.
Figure 2. Mesh example.
We will give results for two kinds of solutions: a Gaussian bump centred on the top side of the unit square Ω Ω Omega\OmegaΩ, given in Example 2, and a variation of the well-known Hamadard solution given in Example 3.
Example 2. Let the Gaussian bump
u = exp ⁡ ( − ( x − 0.5 ) 2 2 σ x − ( y − 1 ) 2 2 σ y ) , σ x = 0.01 , σ y = 0.1 , u = exp ⁡ − ( x − 0.5 ) 2 2 σ x − ( y − 1 ) 2 2 σ y , σ x = 0.01 , σ y = 0.1 , u=exp(-((x-0.5)^(2))/(2sigma_(x))-((y-1)^(2))/(2sigma_(y))),quadsigma_(x)=0.01,sigma_(y)=0.1,u=\exp \left(-\frac{(x-0.5)^{2}}{2 \sigma_{x}}-\frac{(y-1)^{2}}{2 \sigma_{y}}\right), \quad \sigma_{x}=0.01, \sigma_{y}=0.1,u=exp⁡(−(x−0.5)22σx−(y−1)22σy),σx=0.01,σy=0.1,
be a non-homogeneous solution of (14), i.e. f = − Δ u − k 2 u f = − Δ u − k 2 u f=-Delta u-k^(2)uf=-\Delta u-k^{2} uf=−Δu−k2u and q = u | ω q = u ω q=u|_(omega)q=\left.u\right|_{\omega}q=u|ω.
Figure 4a shows that for Example 2, when k = 10 k = 10 k=10k=10k=10, the numerical results strongly agree with the convergence rates expected from Theorem 1 and Theorem 2, and Lemma 4, i.e. sub-linear convergence for the relative error in the L 2 L 2 L^(2)L^{2}L2 - and H 1 H 1 H^(1)H^{1}H1-norms, and quadratic convergence for J ( u h , u h ) J u h , u h J(u_(h),u_(h))\mathcal{J}\left(u_{h}, u_{h}\right)J(uh,uh). Although in Figure 4b we do obtain smaller errors and better than expected convergence rates when k = 50 k = 50 k=50k=50k=50, various numerical experiments indicate that this example's behaviour when increasing the wave number k k kkk is rather a particular one. For oscillatory solutions, such as those in Example 3, with fixed n n nnn, or the homogeneous u = sin ⁡ ( k x / 2 ) cos ⁡ ( k y / 2 ) u = sin ⁡ ( k x / 2 ) cos ⁡ ( k y / 2 ) u=sin(kx//sqrt2)cos(ky//sqrt2)u=\sin (k x / \sqrt{2}) \cos (k y / \sqrt{2})u=sin⁡(kx/2)cos⁡(ky/2), we have noticed that the stability deteriorates when increasing k k kkk.
In the non-convex setting we let
(25) ω = ( 0.25 , 0.75 ) × ( 0 , 0.5 ) , B = ( 0.125 , 0.875 ) × ( 0 , 0.95 ) , (25) ω = ( 0.25 , 0.75 ) × ( 0 , 0.5 ) , B = ( 0.125 , 0.875 ) × ( 0 , 0.95 ) , {:(25)omega=(0.25","0.75)xx(0","0.5)","quad B=(0.125","0.875)xx(0","0.95)",":}\begin{equation*} \omega=(0.25,0.75) \times(0,0.5), \quad B=(0.125,0.875) \times(0,0.95), \tag{25} \end{equation*}(25)ω=(0.25,0.75)×(0,0.5),B=(0.125,0.875)×(0,0.95),
and the concentric disks
(26) ω = D ( ( 0.5 , 0.5 ) , 0.25 ) , B = D ( ( 0.5 , 0.5 ) , 0.45 ) (26) ω = D ( ( 0.5 , 0.5 ) , 0.25 ) , B = D ( ( 0.5 , 0.5 ) , 0.45 ) {:(26)omega=D((0.5","0.5)","0.25)","quad B=D((0.5","0.5)","0.45):}\begin{equation*} \omega=D((0.5,0.5), 0.25), \quad B=D((0.5,0.5), 0.45) \tag{26} \end{equation*}(26)ω=D((0.5,0.5),0.25),B=D((0.5,0.5),0.45)
respectively shown in Figure 3b and Figure 3c, and we notice from Figure 5that the stability strongly deteriorates when one continues the solution outside the convex hull of ω ω omega\omegaω, as the error sizes and rates worsen.
We test the data perturbations by polluting f f fff and q q qqq in (14) with uniformly distributed values in [ − h , h ] [ − h , h ] [-h,h][-h, h][−h,h], respectively [ − h 2 , h 2 ] − h 2 , h 2 [-h^(2),h^(2)]\left[-h^{2}, h^{2}\right][−h2,h2], on every node of the mesh. It can be seen in Figure 6 that the perturbations are visible for an O ( h ) O ( h ) O(h)O(h)O(h) amplitude, but not for an O ( h 2 ) O h 2 O(h^(2))O\left(h^{2}\right)O(h2) one.
Let us recall that the stability estimates for the unique continuation problem are closely related to those for the notoriously ill-posed Cauchy problem, see e.g. [1] or [17]. It is of
Figure 3. Computational domains for Example 2.
Figure 4. Convergence in B B BBB for Example 2 in the convex direction (24).
Figure 5. Convergence in B B BBB for Example 2, k = 10 k = 10 k=10k=10k=10.
Figure 6. Convergence in B B BBB when perturbing f f fff and q q qqq in Example 2 for (24), k = 10 k = 10 k=10k=10k=10.
interest to consider the following variation of a well-known example due to Hadamard, since this example can be used to show that conditional Hölder stability is optimal for the unique continuation problem.
Example 3. Let n ∈ N n ∈ N n inNn \in \mathbb{N}n∈N and consider the Cauchy problem
{ Δ u + k 2 u = 0 in Ω = ( 0 , π ) × ( 0 , 1 ) , u ( x , 0 ) = 0 for x ∈ [ 0 , π ] , u y ( x , 0 ) = sin ⁡ ( n x ) for x ∈ [ 0 , π ] , Δ u + k 2 u = 0       in  Ω = ( 0 , π ) × ( 0 , 1 ) , u ( x , 0 ) = 0       for  x ∈ [ 0 , π ] , u y ( x , 0 ) = sin ⁡ ( n x )       for  x ∈ [ 0 , π ] , {[Delta u+k^(2)u=0," in "Omega=(0","pi)xx(0","1)","],[u(x","0)=0," for "x in[0","pi]","],[u_(y)(x","0)=sin(nx)," for "x in[0","pi]","]:}\begin{cases}\Delta u+k^{2} u=0 & \text { in } \Omega=(0, \pi) \times(0,1), \\ u(x, 0)=0 & \text { for } x \in[0, \pi], \\ u_{y}(x, 0)=\sin (n x) & \text { for } x \in[0, \pi],\end{cases}{Δu+k2u=0 in Ω=(0,π)×(0,1),u(x,0)=0 for x∈[0,π],uy(x,0)=sin⁡(nx) for x∈[0,π],
whose solution for n > k n > k n > kn>kn>k is given by u = 1 n 2 − k 2 sin ⁡ ( n x ) sinh ⁡ ( n 2 − k 2 y ) u = 1 n 2 − k 2 sin ⁡ ( n x ) sinh ⁡ n 2 − k 2 y u=(1)/(sqrt(n^(2)-k^(2)))sin(nx)sinh(sqrt(n^(2)-k^(2))y)u=\frac{1}{\sqrt{n^{2}-k^{2}}} \sin (n x) \sinh \left(\sqrt{n^{2}-k^{2}} y\right)u=1n2−k2sin⁡(nx)sinh⁡(n2−k2y), for n = k n = k n=kn=kn=k by u = sin ⁡ ( k x ) y u = sin ⁡ ( k x ) y u=sin(kx)yu=\sin (k x) yu=sin⁡(kx)y, and for n < k n < k n < kn<kn<k by u = 1 k 2 − n 2 sin ⁡ ( n x ) sin ⁡ ( k 2 − n 2 y ) u = 1 k 2 − n 2 sin ⁡ ( n x ) sin ⁡ k 2 − n 2 y u=(1)/(sqrt(k^(2)-n^(2)))sin(nx)sin(sqrt(k^(2)-n^(2))y)u=\frac{1}{\sqrt{k^{2}-n^{2}}} \sin (n x) \sin \left(\sqrt{k^{2}-n^{2}} y\right)u=1k2−n2sin⁡(nx)sin⁡(k2−n2y).
It can be seen in Figure 7a that the convergence rates agree with the ones predicted for the convex setting
(27) ω = Ω ∖ [ π / 4 , 3 π / 4 ] × [ 0 , 0.25 ] , B = Ω ∖ [ π / 4 , 3 π / 4 ] × [ 0 , 0.95 ] , (27) ω = Ω ∖ [ π / 4 , 3 π / 4 ] × [ 0 , 0.25 ] , B = Ω ∖ [ π / 4 , 3 π / 4 ] × [ 0 , 0.95 ] , {:(27)omega=Omega\\[pi//4","3pi//4]xx[0","0.25]","quad B=Omega\\[pi//4","3pi//4]xx[0","0.95]",":}\begin{equation*} \omega=\Omega \backslash[\pi / 4,3 \pi / 4] \times[0,0.25], \quad B=\Omega \backslash[\pi / 4,3 \pi / 4] \times[0,0.95], \tag{27} \end{equation*}(27)ω=Ω∖[π/4,3π/4]×[0,0.25],B=Ω∖[π/4,3π/4]×[0,0.95],
i.e. sub-linear convergence for the relative error in the L 2 L 2 L^(2)L^{2}L2 - and H 1 H 1 H^(1)H^{1}H1-norms, and quadratic convergence for J ( u h , u h ) J u h , u h J(u_(h),u_(h))\mathcal{J}\left(u_{h}, u_{h}\right)J(uh,uh), although one can notice that the values of the jump stabilizer J ( u h , u h ) J u h , u h J(u_(h),u_(h))\mathcal{J}\left(u_{h}, u_{h}\right)J(uh,uh) visibly increase compared to Example 2.
When continuing the solution in the non-convex direction, the stability strongly deteriorates and for coarse meshes the numerical approximation doesn't reach the convergence regime, as it can be seen in Figure 7b for the non-convex setting
(28) ω = ( π / 4 , 3 π / 4 ) × ( 0 , 0.5 ) , B = ( π / 8 , 7 π / 8 ) × ( 0 , 0.95 ) . (28) ω = ( π / 4 , 3 π / 4 ) × ( 0 , 0.5 ) , B = ( π / 8 , 7 π / 8 ) × ( 0 , 0.95 ) . {:(28)omega=(pi//4","3pi//4)xx(0","0.5)","quad B=(pi//8","7pi//8)xx(0","0.95).:}\begin{equation*} \omega=(\pi / 4,3 \pi / 4) \times(0,0.5), \quad B=(\pi / 8,7 \pi / 8) \times(0,0.95) . \tag{28} \end{equation*}(28)ω=(π/4,3π/4)×(0,0.5),B=(π/8,7π/8)×(0,0.95).

Appendix A.

Example 4. Consider the geometry Ω = ( 0 , 1 ) 2 , ω = ( 0 , 1 ) × ( 0 , ϵ ) Ω = ( 0 , 1 ) 2 , ω = ( 0 , 1 ) × ( 0 , ϵ ) Omega=(0,1)^(2),omega=(0,1)xx(0,epsilon)\Omega=(0,1)^{2}, \omega=(0,1) \times(0, \epsilon)Ω=(0,1)2,ω=(0,1)×(0,ϵ) and B = ( 0 , 1 ) × ( 0 , 1 − ϵ ) B = ( 0 , 1 ) × ( 0 , 1 − ϵ ) B=(0,1)xx(0,1-epsilon)B=(0,1) \times(0,1-\epsilon)B=(0,1)×(0,1−ϵ), and the ansatz u ( x , y ) = e i k x a ( x , y ) u ( x , y ) = e i k x a ( x , y ) u(x,y)=e^(ikx)a(x,y)u(x, y)=e^{i k x} a(x, y)u(x,y)=eikxa(x,y). Let n ∈ N n ∈ N n inNn \in \mathbb{N}n∈N and a ( x , y ) = a 0 ( x , y ) + k − 1 a − 1 ( x , y ) + … + a ( x , y ) = a 0 ( x , y ) + k − 1 a − 1 ( x , y ) + … + a(x,y)=a_(0)(x,y)+k^(-1)a_(-1)(x,y)+dots+a(x, y)=a_{0}(x, y)+k^{-1} a_{-1}(x, y)+\ldots+a(x,y)=a0(x,y)+k−1a−1(x,y)+…+

(a) Convex direction (27). Circles: H 1 H 1 H^(1)H^{1}H1-error, rate ≈ 0.94 ≈ 0.94 ~~0.94\approx 0.94≈0.94; squares: L 2 L 2 L^(2)L^{2}L2-error, rate ≈ 0.83 ≈ 0.83 ~~0.83\approx 0.83≈0.83; down triangles: h − 1 J ( u h , u h ) h − 1 J u h , u h h^(-1)J(u_(h),u_(h))h^{-1} \mathcal{J}\left(u_{h}, u_{h}\right)h−1J(uh,uh), rate ≈ 1 ≈ 1 ~~1\approx 1≈1; up triangles: ‖ z ‖ W ‖ z ‖ W ||z||_(W)\|z\|_{W}‖z‖W, rate ≈ 1.6 ≈ 1.6 ~~1.6\approx 1.6≈1.6.

(b) Non-convex direction (28). Circles: H 1 H 1 H^(1)H^{1}H1 error; squares: L 2 L 2 L^(2)L^{2}L2-error; down triangles: h − 1 J ( u h , u h ) h − 1 J u h , u h h^(-1)J(u_(h),u_(h))h^{-1} \mathcal{J}\left(u_{h}, u_{h}\right)h−1J(uh,uh); up triangles: ‖ z ‖ W ‖ z ‖ W ||z||_(W)\|z\|_{W}‖z‖W.
Figure 7. Convergence in B B BBB for Example 3, k = 10 , n = 12 k = 10 , n = 12 k=10,n=12k=10, n=12k=10,n=12.
k − n a − n ( x , y ) k − n a − n ( x , y ) k^(-n)a_(-n)(x,y)k^{-n} a_{-n}(x, y)k−na−n(x,y). We have that
Δ u + k 2 u = e i k x ( 2 i k ∂ x a + Δ a ) Δ u + k 2 u = e i k x 2 i k ∂ x a + Δ a Delta u+k^(2)u=e^(ikx)(2ikdel_(x)a+Delta a)\Delta u+k^{2} u=e^{i k x}\left(2 i k \partial_{x} a+\Delta a\right)Δu+k2u=eikx(2ik∂xa+Δa)
and we choose a j , j = 0 , … , − n a j , j = 0 , … , − n a_(j),j=0,dots,-na_{j}, j=0, \ldots,-naj,j=0,…,−n such that
(29) ∂ x a 0 = 0 , 2 i ∂ x a j + Δ a j + 1 = 0 , − j = 1 , … , n (29) ∂ x a 0 = 0 , 2 i ∂ x a j + Δ a j + 1 = 0 , − j = 1 , … , n {:(29)del_(x)a_(0)=0","quad2idel_(x)a_(j)+Deltaa_(j+1)=0","quad-j=1","dots","n:}\begin{equation*} \partial_{x} a_{0}=0, \quad 2 i \partial_{x} a_{j}+\Delta a_{j+1}=0, \quad-j=1, \ldots, n \tag{29} \end{equation*}(29)∂xa0=0,2i∂xaj+Δaj+1=0,−j=1,…,n
Then
Δ u + k 2 u = e i k x k − n Δ a − n Δ u + k 2 u = e i k x k − n Δ a − n Delta u+k^(2)u=e^(ikx)k^(-n)Deltaa_(-n)\Delta u+k^{2} u=e^{i k x} k^{-n} \Delta a_{-n}Δu+k2u=eikxk−nΔa−n
and ‖ Δ u + k 2 u ‖ L 2 ( Ω ) = k − n ‖ Δ a − N ‖ L 2 ( Ω ) Δ u + k 2 u L 2 ( Ω ) = k − n Δ a − N L 2 ( Ω ) ||Delta u+k^(2)u||_(L^(2)(Omega))=k^(-n)||Deltaa_(-N)||_(L^(2)(Omega))\left\|\Delta u+k^{2} u\right\|_{L^{2}(\Omega)}=k^{-n}\left\|\Delta a_{-N}\right\|_{L^{2}(\Omega)}‖Δu+k2u‖L2(Ω)=k−n‖Δa−N‖L2(Ω). Since a j , j = 0 , … , − n a j , j = 0 , … , − n a_(j),j=0,dots,-na_{j}, j=0, \ldots,-naj,j=0,…,−n, are independent of k k kkk we obtain
‖ Δ u + k 2 u ‖ L 2 ( Ω ) = C k − n Δ u + k 2 u L 2 ( Ω ) = C k − n ||Delta u+k^(2)u||_(L^(2)(Omega))=Ck^(-n)\left\|\Delta u+k^{2} u\right\|_{L^{2}(\Omega)}=C k^{-n}‖Δu+k2u‖L2(Ω)=Ck−n
We can solve (29) such that a 0 ( x , y ) = a 0 ( y ) , supp ( a 0 ) ⊂ ( ϵ , 1 − ϵ ) a 0 ( x , y ) = a 0 ( y ) , supp a 0 ⊂ ( ϵ , 1 − ϵ ) a_(0)(x,y)=a_(0)(y),supp(a_(0))sub(epsilon,1-epsilon)a_{0}(x, y)=a_{0}(y), \operatorname{supp}\left(a_{0}\right) \subset(\epsilon, 1-\epsilon)a0(x,y)=a0(y),supp(a0)⊂(ϵ,1−ϵ) and supp ( a ) ⊂ [ 0 , 1 ] × ( ϵ , 1 − ϵ ) supp ( a ) ⊂ [ 0 , 1 ] × ( ϵ , 1 − ϵ ) supp(a)sub[0,1]xx(epsilon,1-epsilon)\operatorname{supp}(a) \subset[0,1] \times (\epsilon, 1-\epsilon)supp(a)⊂[0,1]×(ϵ,1−ϵ). Then
u | ω = 0 , and ‖ u ‖ H 1 ( B ) = ‖ u ‖ H 1 ( Ω ) = C k , for large k . u ω = 0 ,  and  ‖ u ‖ H 1 ( B ) = ‖ u ‖ H 1 ( Ω ) = C k ,  for large  k . u|_(omega)=0,quad" and "||u||_(H^(1)(B))=||u||_(H^(1)(Omega))=Ck,quad" for large "k.\left.u\right|_{\omega}=0, \quad \text { and }\|u\|_{H^{1}(B)}=\|u\|_{H^{1}(\Omega)}=C k, \quad \text { for large } k .u|ω=0, and ‖u‖H1(B)=‖u‖H1(Ω)=Ck, for large k.
The estimate (2) then becomes
k ≤ C k − α n k 1 − α , i.e. k α ( n + 1 ) ≤ C k ≤ C k − α n k 1 − α ,  i.e.  k α ( n + 1 ) ≤ C k <= Ck^(-alpha n)k^(1-alpha),quad" i.e. "k^(alpha(n+1)) <= Ck \leq C k^{-\alpha n} k^{1-\alpha}, \quad \text { i.e. } k^{\alpha(n+1)} \leq Ck≤Ck−αnk1−α, i.e. kα(n+1)≤C
Choosing large n n nnn we see that C C CCC depends on k k kkk, and for any N ∈ N , C ≤ k N N ∈ N , C ≤ k N N inN,C <= k^(N)N \in \mathbb{N}, C \leq k^{N}N∈N,C≤kN cannot hold.
Proof of Lemma 1. Recall the following identities for a function w w www and vector fields X X XXX and Y Y YYY
div ( w X ) = ( ∇ w , X ) + w div X , D 2 w ( X , Y ) = ( D X ∇ w , Y ) div ( w X ) = ( ∇ w , X ) + w div X , D 2 w ( X , Y ) = D X ∇ w , Y div(wX)=(grad w,X)+w div X,quadD^(2)w(X,Y)=(D_(X)grad w,Y)\operatorname{div}(w X)=(\nabla w, X)+w \operatorname{div} X, \quad D^{2} w(X, Y)=\left(D_{X} \nabla w, Y\right)div(wX)=(∇w,X)+wdivX,D2w(X,Y)=(DX∇w,Y)
where D X D X D_(X)D_{X}DX is the covariant derivative. Recall also that the Hessian is symmetric, i.e. D 2 w ( X , Y ) = D 2 w ( Y , X ) D 2 w ( X , Y ) = D 2 w ( Y , X ) D^(2)w(X,Y)=D^(2)w(Y,X)D^{2} w(X, Y)= D^{2} w(Y, X)D2w(X,Y)=D2w(Y,X). We have
e ℓ Δ w = Δ v + b + ( q − k 2 ) v = Δ v − σ v − 2 ( ∇ v , ∇ ℓ ) + σ v − ( Δ ℓ ) v + | ∇ ℓ | 2 v e ℓ Δ w = Δ v + b + q − k 2 v = Δ v − σ v − 2 ( ∇ v , ∇ ℓ ) + σ v − ( Δ ℓ ) v + | ∇ ℓ | 2 v e^(ℓ)Delta w=Delta v+b+(q-k^(2))v=Delta v-sigma v-2(grad v,gradℓ)+sigma v-(Deltaℓ)v+|gradℓ|^(2)ve^{\ell} \Delta w=\Delta v+b+\left(q-k^{2}\right) v=\Delta v-\sigma v-2(\nabla v, \nabla \ell)+\sigma v-(\Delta \ell) v+|\nabla \ell|^{2} veℓΔw=Δv+b+(q−k2)v=Δv−σv−2(∇v,∇ℓ)+σv−(Δℓ)v+|∇ℓ|2v
Indeed
Δ v = div ( ∇ ( e ℓ w ) ) = div ( v ∇ ℓ + e ℓ ∇ w ) = ( ∇ v , ∇ ℓ ) + v Δ ℓ + ( ∇ e ℓ , ∇ w ) + e ℓ Δ w = 2 ( ∇ v , ∇ ℓ ) + ( Δ ℓ − | ∇ ℓ | 2 ) v + e ℓ Δ w Δ v = div ∇ e ℓ w = div v ∇ ℓ + e ℓ ∇ w = ( ∇ v , ∇ ℓ ) + v Δ ℓ + ∇ e ℓ , ∇ w + e ℓ Δ w = 2 ( ∇ v , ∇ ℓ ) + Δ ℓ − | ∇ ℓ | 2 v + e ℓ Δ w {:[Delta v=div(grad(e^(ℓ)w))=div(v gradℓ+e^(ℓ)grad w)],[=(grad v","gradℓ)+v Deltaℓ+(grade^(ℓ),grad w)+e^(ℓ)Delta w],[=2(grad v","gradℓ)+(Deltaℓ-|gradℓ|^(2))v+e^(ℓ)Delta w]:}\begin{aligned} \Delta v & =\operatorname{div}\left(\nabla\left(e^{\ell} w\right)\right)=\operatorname{div}\left(v \nabla \ell+e^{\ell} \nabla w\right) \\ & =(\nabla v, \nabla \ell)+v \Delta \ell+\left(\nabla e^{\ell}, \nabla w\right)+e^{\ell} \Delta w \\ & =2(\nabla v, \nabla \ell)+\left(\Delta \ell-|\nabla \ell|^{2}\right) v+e^{\ell} \Delta w \end{aligned}Δv=div(∇(eℓw))=div(v∇ℓ+eℓ∇w)=(∇v,∇ℓ)+vΔℓ+(∇eℓ,∇w)+eℓΔw=2(∇v,∇ℓ)+(Δℓ−|∇ℓ|2)v+eℓΔw
where we have used the identity
( ∇ e ℓ , ∇ w ) = ( e ℓ ∇ ℓ , ∇ w ) = ( ∇ ℓ , ∇ ( e ℓ w ) ) − ( ∇ ℓ , w ∇ e ℓ ) = ( ∇ ℓ , ∇ v ) − v | ∇ ℓ | 2 ∇ e ℓ , ∇ w = e ℓ ∇ ℓ , ∇ w = ∇ ℓ , ∇ e ℓ w − ∇ ℓ , w ∇ e ℓ = ( ∇ ℓ , ∇ v ) − v | ∇ ℓ | 2 {:[(grade^(ℓ),grad w)=(e^(ℓ)gradℓ,grad w)=(gradℓ,grad(e^(ℓ)w))-(gradℓ,w grade^(ℓ))],[=(gradℓ","grad v)-v|gradℓ|^(2)]:}\begin{aligned} \left(\nabla e^{\ell}, \nabla w\right) & =\left(e^{\ell} \nabla \ell, \nabla w\right)=\left(\nabla \ell, \nabla\left(e^{\ell} w\right)\right)-\left(\nabla \ell, w \nabla e^{\ell}\right) \\ & =(\nabla \ell, \nabla v)-v|\nabla \ell|^{2} \end{aligned}(∇eℓ,∇w)=(eℓ∇ℓ,∇w)=(∇ℓ,∇(eℓw))−(∇ℓ,w∇eℓ)=(∇ℓ,∇v)−v|∇ℓ|2
Thus
(30) e 2 ℓ ( Δ w + k 2 w ) 2 / 2 = ( Δ v + b + q v ) 2 / 2 = ( Δ v + q v ) 2 / 2 + b 2 / 2 + b Δ v + b q v (30) e 2 ℓ Δ w + k 2 w 2 / 2 = ( Δ v + b + q v ) 2 / 2 = ( Δ v + q v ) 2 / 2 + b 2 / 2 + b Δ v + b q v {:(30)e^(2ℓ)(Delta w+k^(2)w)^(2)//2=(Delta v+b+qv)^(2)//2=(Delta v+qv)^(2)//2+b^(2)//2+b Delta v+bqv:}\begin{equation*} e^{2 \ell}\left(\Delta w+k^{2} w\right)^{2} / 2=(\Delta v+b+q v)^{2} / 2=(\Delta v+q v)^{2} / 2+b^{2} / 2+b \Delta v+b q v \tag{30} \end{equation*}(30)e2ℓ(Δw+k2w)2/2=(Δv+b+qv)2/2=(Δv+qv)2/2+b2/2+bΔv+bqv
and it remains to study the cross terms b Δ v b Δ v b Delta vb \Delta vbΔv and b q v b q v bqvb q vbqv.
Let us begin by studying β Δ v β Δ v beta Delta v\beta \Delta vβΔv where β = − 2 ( ∇ v , ∇ ℓ ) β = − 2 ( ∇ v , ∇ ℓ ) beta=-2(grad v,gradℓ)\beta=-2(\nabla v, \nabla \ell)β=−2(∇v,∇ℓ). We have
β Δ v = div ( β ∇ v ) − ( ∇ β , ∇ v ) β Δ v = div ( β ∇ v ) − ( ∇ β , ∇ v ) beta Delta v=div(beta grad v)-(grad beta,grad v)\beta \Delta v=\operatorname{div}(\beta \nabla v)-(\nabla \beta, \nabla v)βΔv=div(β∇v)−(∇β,∇v)
and
− ( ∇ β , ∇ v ) = 2 ( ∇ ( ∇ v , ∇ ℓ ) , ∇ v ) = 2 ( D ∇ v ∇ v , ∇ ℓ ) + 2 ( ∇ v , D ∇ v ∇ ℓ ) = 2 D 2 v ( ∇ v , ∇ ℓ ) + 2 D 2 ℓ ( ∇ v , ∇ v ) − ( ∇ β , ∇ v ) = 2 ( ∇ ( ∇ v , ∇ ℓ ) , ∇ v ) = 2 D ∇ v ∇ v , ∇ ℓ + 2 ∇ v , D ∇ v ∇ ℓ = 2 D 2 v ( ∇ v , ∇ ℓ ) + 2 D 2 ℓ ( ∇ v , ∇ v ) {:[-(grad beta","grad v)=2(grad(grad v","gradℓ)","grad v)=2(D_(grad v)grad v,gradℓ)+2(grad v,D_(grad v)gradℓ)],[=2D^(2)v(grad v","gradℓ)+2D^(2)ℓ(grad v","grad v)]:}\begin{aligned} -(\nabla \beta, \nabla v) & =2(\nabla(\nabla v, \nabla \ell), \nabla v)=2\left(D_{\nabla v} \nabla v, \nabla \ell\right)+2\left(\nabla v, D_{\nabla v} \nabla \ell\right) \\ & =2 D^{2} v(\nabla v, \nabla \ell)+2 D^{2} \ell(\nabla v, \nabla v) \end{aligned}−(∇β,∇v)=2(∇(∇v,∇ℓ),∇v)=2(D∇v∇v,∇ℓ)+2(∇v,D∇v∇ℓ)=2D2v(∇v,∇ℓ)+2D2ℓ(∇v,∇v)
Finally
(31) 2 D 2 v ( ∇ v , ∇ ℓ ) = 2 D 2 v ( ∇ ℓ , ∇ v ) = 2 ( D ∇ ℓ ∇ v , ∇ v ) = ( ∇ ℓ , ∇ | ∇ v | 2 ) = div ( | ∇ v | 2 ∇ ℓ ) − | ∇ v | 2 Δ ℓ (31) 2 D 2 v ( ∇ v , ∇ ℓ ) = 2 D 2 v ( ∇ ℓ , ∇ v ) = 2 D ∇ ℓ ∇ v , ∇ v = ∇ ℓ , ∇ | ∇ v | 2 = div | ∇ v | 2 ∇ ℓ − | ∇ v | 2 Δ ℓ {:[(31)2D^(2)v(grad v","gradℓ)=2D^(2)v(gradℓ","grad v)=2(D_(gradℓ)grad v,grad v)=(gradℓ,grad|grad v|^(2))],[=div(|grad v|^(2)gradℓ)-|grad v|^(2)Deltaℓ]:}\begin{align*} 2 D^{2} v(\nabla v, \nabla \ell) & =2 D^{2} v(\nabla \ell, \nabla v)=2\left(D_{\nabla \ell} \nabla v, \nabla v\right)=\left(\nabla \ell, \nabla|\nabla v|^{2}\right) \tag{31}\\ & =\operatorname{div}\left(|\nabla v|^{2} \nabla \ell\right)-|\nabla v|^{2} \Delta \ell \end{align*}(31)2D2v(∇v,∇ℓ)=2D2v(∇ℓ,∇v)=2(D∇ℓ∇v,∇v)=(∇ℓ,∇|∇v|2)=div(|∇v|2∇ℓ)−|∇v|2Δℓ
To summarize, for β = − 2 ( ∇ v , ∇ ℓ ) β = − 2 ( ∇ v , ∇ ℓ ) beta=-2(grad v,gradℓ)\beta=-2(\nabla v, \nabla \ell)β=−2(∇v,∇ℓ) it holds that
(32) β Δ v = − Δ ℓ | ∇ v | 2 + 2 D 2 ℓ ( ∇ v , ∇ v ) + div ( β ∇ v + | ∇ v | 2 ∇ ℓ ) (32) β Δ v = − Δ ℓ | ∇ v | 2 + 2 D 2 ℓ ( ∇ v , ∇ v ) + div β ∇ v + | ∇ v | 2 ∇ ℓ {:(32)beta Delta v=-Deltaℓ|grad v|^(2)+2D^(2)ℓ(grad v","grad v)+div(beta grad v+|grad v|^(2)gradℓ):}\begin{equation*} \beta \Delta v=-\Delta \ell|\nabla v|^{2}+2 D^{2} \ell(\nabla v, \nabla v)+\operatorname{div}\left(\beta \nabla v+|\nabla v|^{2} \nabla \ell\right) \tag{32} \end{equation*}(32)βΔv=−Δℓ|∇v|2+2D2ℓ(∇v,∇v)+div(β∇v+|∇v|2∇ℓ)
Consider now β Δ v β Δ v beta Delta v\beta \Delta vβΔv where β = − σ v β = − σ v beta=-sigma v\beta=-\sigma vβ=−σv. We have
− ( ∇ β , ∇ v ) = ( ∇ σ , ∇ v ) v + σ | ∇ v | 2 − ( ∇ β , ∇ v ) = ( ∇ σ , ∇ v ) v + σ | ∇ v | 2 -(grad beta,grad v)=(grad sigma,grad v)v+sigma|grad v|^(2)-(\nabla \beta, \nabla v)=(\nabla \sigma, \nabla v) v+\sigma|\nabla v|^{2}−(∇β,∇v)=(∇σ,∇v)v+σ|∇v|2
whence for β = − σ v β = − σ v beta=-sigma v\beta=-\sigma vβ=−σv it holds that
(33) β Δ v = σ | ∇ v | 2 + div ( β ∇ v ) + ( ∇ σ , ∇ v ) v (33) β Δ v = σ | ∇ v | 2 + div ( β ∇ v ) + ( ∇ σ , ∇ v ) v {:(33)beta Delta v=sigma|grad v|^(2)+div(beta grad v)+(grad sigma","grad v)v:}\begin{equation*} \beta \Delta v=\sigma|\nabla v|^{2}+\operatorname{div}(\beta \nabla v)+(\nabla \sigma, \nabla v) v \tag{33} \end{equation*}(33)βΔv=σ|∇v|2+div(β∇v)+(∇σ,∇v)v
Now (32) and (33) imply
(34) b Δ v = a | ∇ v | 2 + 2 D 2 ℓ ( ∇ v , ∇ v ) + div ( b ∇ v + c 0 ) + R 0 (34) b Δ v = a | ∇ v | 2 + 2 D 2 ℓ ( ∇ v , ∇ v ) + div b ∇ v + c 0 + R 0 {:(34)b Delta v=a|grad v|^(2)+2D^(2)ℓ(grad v","grad v)+div(b grad v+c_(0))+R_(0):}\begin{equation*} b \Delta v=a|\nabla v|^{2}+2 D^{2} \ell(\nabla v, \nabla v)+\operatorname{div}\left(b \nabla v+c_{0}\right)+R_{0} \tag{34} \end{equation*}(34)bΔv=a|∇v|2+2D2ℓ(∇v,∇v)+div(b∇v+c0)+R0
where c 0 = | ∇ v | 2 ∇ ℓ c 0 = | ∇ v | 2 ∇ ℓ c_(0)=|grad v|^(2)gradℓc_{0}=|\nabla v|^{2} \nabla \ellc0=|∇v|2∇ℓ and R 0 = ( ∇ σ , ∇ v ) v R 0 = ( ∇ σ , ∇ v ) v R_(0)=(grad sigma,grad v)vR_{0}=(\nabla \sigma, \nabla v) vR0=(∇σ,∇v)v.
Let us now study the second cross term in (30). We have
− 2 ( ∇ v , ∇ ℓ ) q v = − ( ∇ v 2 , q ∇ ℓ ) = v 2 div ( q ∇ ℓ ) − div ( v 2 q ∇ ℓ ) − 2 ( ∇ v , ∇ ℓ ) q v = − ∇ v 2 , q ∇ ℓ = v 2 div ( q ∇ ℓ ) − div v 2 q ∇ ℓ -2(grad v,gradℓ)qv=-(gradv^(2),q gradℓ)=v^(2)div(q gradℓ)-div(v^(2)q gradℓ)-2(\nabla v, \nabla \ell) q v=-\left(\nabla v^{2}, q \nabla \ell\right)=v^{2} \operatorname{div}(q \nabla \ell)-\operatorname{div}\left(v^{2} q \nabla \ell\right)−2(∇v,∇ℓ)qv=−(∇v2,q∇ℓ)=v2div(q∇ℓ)−div(v2q∇ℓ)
whence, recalling that q = k 2 + a + | ∇ ℓ | 2 q = k 2 + a + | ∇ ℓ | 2 q=k^(2)+a+|gradℓ|^(2)q=k^{2}+a+|\nabla \ell|^{2}q=k2+a+|∇ℓ|2 and − a = − σ + Δ ℓ − a = − σ + Δ ℓ -a=-sigma+Deltaℓ-a=-\sigma+\Delta \ell−a=−σ+Δℓ,
(35) b q v = ( − σ q + div ( q ∇ ℓ ) ) v 2 + div c 1 = ( − | ∇ ℓ | 2 σ + div ( | ∇ ℓ | 2 ∇ ℓ ) ) v 2 − k 2 a v 2 + div c 1 + R 1 (35) b q v = ( − σ q + div ( q ∇ ℓ ) ) v 2 + div c 1 = − | ∇ ℓ | 2 σ + div | ∇ ℓ | 2 ∇ ℓ v 2 − k 2 a v 2 + div c 1 + R 1 {:[(35)bqv=(-sigma q+div(q gradℓ))v^(2)+divc_(1)],[=(-|gradℓ|^(2)sigma+div(|gradℓ|^(2)gradℓ))v^(2)-k^(2)av^(2)+divc_(1)+R_(1)]:}\begin{align*} b q v & =(-\sigma q+\operatorname{div}(q \nabla \ell)) v^{2}+\operatorname{div} c_{1} \tag{35}\\ & =\left(-|\nabla \ell|^{2} \sigma+\operatorname{div}\left(|\nabla \ell|^{2} \nabla \ell\right)\right) v^{2}-k^{2} a v^{2}+\operatorname{div} c_{1}+R_{1} \end{align*}(35)bqv=(−σq+div(q∇ℓ))v2+divc1=(−|∇ℓ|2σ+div(|∇ℓ|2∇ℓ))v2−k2av2+divc1+R1
where c 1 = − q v 2 ∇ ℓ c 1 = − q v 2 ∇ ℓ c_(1)=-qv^(2)gradℓc_{1}=-q v^{2} \nabla \ellc1=−qv2∇ℓ and R 1 = ( div ( a ∇ ℓ ) − a σ ) v 2 R 1 = ( div ( a ∇ ℓ ) − a σ ) v 2 R_(1)=(div(a gradℓ)-a sigma)v^(2)R_{1}=(\operatorname{div}(a \nabla \ell)-a \sigma) v^{2}R1=(div(a∇ℓ)−aσ)v2. The identity (31) with v = ℓ v = ℓ v=ℓv=\ellv=ℓ implies that
div ( | ∇ ℓ | 2 ∇ ℓ ) = 2 D 2 ℓ ( ∇ ℓ , ∇ ℓ ) + | ∇ ℓ | 2 Δ ℓ , div | ∇ ℓ | 2 ∇ ℓ = 2 D 2 ℓ ( ∇ ℓ , ∇ ℓ ) + | ∇ ℓ | 2 Δ ℓ , div(|gradℓ|^(2)gradℓ)=2D^(2)ℓ(gradℓ,gradℓ)+|gradℓ|^(2)Deltaℓ,\operatorname{div}\left(|\nabla \ell|^{2} \nabla \ell\right)=2 D^{2} \ell(\nabla \ell, \nabla \ell)+|\nabla \ell|^{2} \Delta \ell,div(|∇ℓ|2∇ℓ)=2D2ℓ(∇ℓ,∇ℓ)+|∇ℓ|2Δℓ,
whence, recalling that σ = a + Δ ℓ σ = a + Δ ℓ sigma=a+Deltaℓ\sigma=a+\Delta \ellσ=a+Δℓ,
(36) − | ∇ ℓ | 2 σ + div ( | ∇ ℓ | 2 ∇ ℓ ) = − a | ∇ ℓ | 2 + 2 D 2 ℓ ( ∇ ℓ , ∇ ℓ ) (36) − | ∇ ℓ | 2 σ + div | ∇ ℓ | 2 ∇ ℓ = − a | ∇ ℓ | 2 + 2 D 2 ℓ ( ∇ ℓ , ∇ ℓ ) {:(36)-|gradℓ|^(2)sigma+div(|gradℓ|^(2)gradℓ)=-a|gradℓ|^(2)+2D^(2)ℓ(gradℓ","gradℓ):}\begin{equation*} -|\nabla \ell|^{2} \sigma+\operatorname{div}\left(|\nabla \ell|^{2} \nabla \ell\right)=-a|\nabla \ell|^{2}+2 D^{2} \ell(\nabla \ell, \nabla \ell) \tag{36} \end{equation*}(36)−|∇ℓ|2σ+div(|∇ℓ|2∇ℓ)=−a|∇ℓ|2+2D2ℓ(∇ℓ,∇ℓ)
The claim follows by combining (36), (35), (34) and (30).
Acknowledgements. Erik Burman was supported by EPSRC grants EP/P01576X/1 and EP/P012434/1. Lauri Oksanen was supported by EPSRC grants EP/L026473/1 and EP/P01593X/1.

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  1. Department of Mathematics, University College London, Gower Street, London UK, WC1E 6BT.
    E-mail addresses: {e.burman, mihai.nechita.16, l.oksanen}@ucl.ac.uk.
    Date: October 29, 2018.
2019

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