The numerical approximation of an inverse problem subject to the convection–diffusion equation when diffusion dominates is studied. We derive Carleman estimates that are on a form suitable for use in numerical analysis and with explicit dependence on the Peclet number. A stabilized finite element method is then proposed and analysed. An upper bound on the condition number is first derived. Combining the stability estimates on the continuous problem with the numerical stability of the method, we then obtain error estimates in local H1- or L2-norms that are optimal with respect to the approximation order, the problem’s stability and perturbations in data. The convergence order is the same for both norms, but the H1-estimate requires an additional divergence assumption for theconvective field. The theory is illustrated in some computational examples
Authors
Erik Burman Department of Mathematics, University College London, Gower Street, London UK, WC1E 6BT
Mihai Nechita Department of Mathematics, University College London, Gower Street, London UK, WC1E 6BT
Lauri Oksanen Department of Mathematics, University College London, Gower Street, London UK, WC1E 6BT
Keywords
convection-diffusion equation; FEM (Finite Element Method);
Paper coordinates
E. Burman, M. Nechita, L. Oksanen, A stabilized finite element method for inverse problems subject to the convection-diffusion equation. I: diffusion-dominated regime, Numer. Math. 144 (2020), pp. 451-477. https://doi.org/10.1007/s00211-019-01087-x
[1] R. Becker and B. Vexler. Optimal control of the convection-diffusion equation using stabilized finite element methods. Numer. Math., 106(3):349–367, 2007.
[2] S. C. Brenner. Poincare-Friedrichs inequalities for piecewise H1 functions. SIAM J. Numer. Anal., 41(1):306–324, 2003.
[3] E. Burman. A unified analysis for conforming and nonconforming stabilized finite element methods using interior penalty. SIAM J. Numer. Anal., 43(5):2012–2033, 2005.
[4] E. Burman. Stabilized finite element methods for nonsymmetric, noncoercive, and ill-posed problems. Part I: elliptic equations. SIAM J. Sci. Comput., 35(6):A2752 A2780, 2013.
[5] E. Burman. Error estimates for stabilized finite element methods applied to ill-posed problems. C. R. Math. Acad. Sci. Paris, 352(7-8):655–659, 2014.
[6] E. Burman, P. Hansbo, and M. G. Larson. Solving ill-posed control problems by stabilized finite element methods: an alternative to Tikhonov regularization. Inverse Problems, 34:035004, 2018.
[7] E. Burman, M. Nechita, and L. Oksanen. A stabilized finite element method for inverse problems subject to the convection–diffusion equation. II: convection dominated regime. in preparation, 2018.
[8] E. Burman, M. Nechita, and L. Oksanen. Unique continuation for the Helmholtz equation using stabilized finite element methods. J. Math. Pures Appl., 2018.
[9] E. Burman and L. Oksanen. Data assimilation for the heat equation using stabilized finite element methods. Numer. Math., 139(3):505–528, 2018.
[10] L. Dede’ and A. Quarteroni. Optimal control and numerical adaptivity for advection-diffusion equations. M2AN Math. Model. Numer. Anal., 39(5):1019–1040, 2005.
[11] D. Dos Santos Ferreira, C. E. Kenig, M. Salo, and G. Uhlmann. Limiting Carleman weights and anisotropic inverse problems. Invent. Math., 178(1):119–171, 2009.
[12] A. Ern and J.-L. Guermond. Theory and practice of finite elements, volume 159 of Applied Mathematical Sciences. Springer-Verlag, New York, 2004.
[13] A. Ern and J.-L. Guermond. Evaluation of the condition number in linear systems arising in finite element approximations. M2AN Math. Model. Numer. Anal., 40(1):29 48, 2006.
[14] F. Hecht. New development in FreeFem++. J. Numer. Math., 20(3-4):251–265, 2012.
[15] M. Hinze, N. Yan, and Z. Zhou. Variational discretization for optimal control governed by convection dominated diffusion equations. J. Comput. Math., 27(2-3):237–253, 2009.
[16] J. Le Rousseau and G. Lebeau. On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations. ESAIM Control Optim. Calc. Var., 18(3):712–747, 2012.
[17] E. Malinnikova and S. Vessella. Quantitative uniqueness for elliptic equations with singular lower order terms. Math. Ann., 353(4):1157–1181, 2012.
[18] P. Monk and E. Suli. The adaptive computation of far-field patterns by a posteriori error estimation of linear functionals. SIAM J. Numer. Anal., 36(1):251–274, 1999.
[19] N. Yan and Z. Zhou. A priori and a posteriori error analysis of edge stabilization Galerkin method for the optimal control problem governed by convection-dominated diffusion equation. J. Comput. Appl. Math., 223(1):198–217, 2009.
[20] M. Zworski. Semiclassical analysis, volume 138 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2012.[32] M.D. Zeiler, ADADELTA : An adaptive learning rate method, 2012,http://arxiv.org/abs/1212.5701.
1811.00431v2
A STABILIZED FINITE ELEMENT METHOD FOR INVERSE PROBLEMS SUBJECT TO THE CONVECTION-DIFFUSION EQUATION. I: DIFFUSION-DOMINATED REGIME
ERIK BURMAN, MIHAI NECHITA, AND LAURI OKSANEN
Abstract
The numerical approximation of an inverse problem subject to the convectiondiffusion equation when diffusion dominates is studied. We derive Carleman estimates that are of a form suitable for use in numerical analysis and with explicit dependence on the Péclet number. A stabilized finite element method is then proposed and analysed. An upper bound on the condition number is first derived. Combining the stability estimates on the continuous problem with the numerical stability of the method, we then obtain error estimates in localH^(1)H^{1}- orL^(2)L^{2}-norms that are optimal with respect to the approximation order, the problem's stability and perturbations in data. The convergence order is the same for both norms, but theH^(1)H^{1}-estimate requires an additional divergence assumption for the convective field. The theory is illustrated in some computational examples.
whereOmega subR^(n)\Omega \subset \mathbb{R}^{n}is open, bounded and connected,mu > 0\mu>0is the diffusion coefficient andbeta in[W^(1,oo)(Omega)]^(n)\beta \in \left[W^{1, \infty}(\Omega)\right]^{n}is the convective velocity field. We assume that no information is given on the boundarydel Omega\partial \Omegaand that there exists a solutionu inH^(2)(Omega)u \in H^{2}(\Omega)satisfying (1). For an open and connected subsetomega sub Omega\omega \subset \Omega, define the perturbed restrictiontilde(U)_(omega):=u∣_(omega)+delta\tilde{U}_{\omega}:=u \mid{ }_{\omega}+\delta, wheredelta inL^(2)(omega)\delta \in L^{2}(\omega)is an unknown function modelling measurement noise. The data assimilation (or unique continuation) problem consists in findinguugivenffandtilde(U)_(omega)\tilde{U}_{\omega}. Here the coefficientsmu\muandbeta\beta, and the source termffare assumed to be known. This linear problem is ill-posed and it is closely related to the elliptic Cauchy problem, see e.g. [ARRV09]. Potential applications include for example flow problems for which full boundary data are not accessible, but where local measurements (in a subset of the domain or on a part of the boundary) can be obtained.
The aim is to design a finite element method for data assimilation with weakly consistent regularization applied to the convection-diffusion equation (1). In the present analysis we consider the regime where diffusion dominates and in the companion paper [BNO19a] we treat the one with dominating convective transport. To make this more precise we introduce the Péclet number associated to a given length scalellby
for a suitable norm|*||\cdot|forbeta\beta. Ifhhdenotes the characteristic length scale of the computation, we define the diffusive regime byPe(h) < 1P e(h)<1and the convective regime byPe(h) > 1P e(h)>1. It is known that the character of the system changes drastically in the two regimes and we therefore need to apply different concepts of stability in the two cases. In the present paper we assume that the Péclet number is small and we use an approach similar to that employed for the Laplace equation in [Bur14], for the Helmholtz equation in [BNO19b] and for the heat equation in [BO18], that is we combine conditional stability estimates for the physical problem with optimal numerical stability obtained using a bespoke weakly consistent stabilizing term. For high Péclet numbers on the other hand, we prove in [BNO19a] weighted estimates directly on the discrete solution, that reflect the anisotropic character of the convection-diffusion problem.
In the case of optimal control problems subject to convection-diffusion problems that are well-posed, there are several works in the literature on stabilized finite element methods. In [DQ05] the authors considered stabilization using a Galerkin least squares approach in the Lagrangian. Symmetric stabilization in the form of local projection stabilization was proposed in [BV07] and using penalty on the gradient jumps in [YZ09, HYZ09]. The key difference between the well-posed case and the ill-posed case that we consider herein is that we can not use stability of neither the forward nor the backward equations. Crucial instead is the convergence of the weakly consistent stabilizing terms and the matching of the quantities in the discrete method and the available (best) stability of the continuous problem. Such considerations lead to results both in the case of high and low Péclet numbers, but the different stability properties in the two regimes lead to a different analysis for each case that will be considered in the two parts of this paper.
The main results of this current work are the convergence estimates with explicit dependence on the Péclet number in Theorem 1 and Theorem 2, that rely on the continuous three-ball inequalities in Lemma 2 and Corollary 2.
2. Stability estimates
We prove conditional stability estimates for the unique continuation problem subject to the convection-diffusion equation (1) in the form of three-ball inequalities, see e.g. [MV12] and the references therein. The novelty here is that we keep track of explicit dependence on the diffusion coefficientmu\muand the convective vector fieldbeta\beta. The first such inequality is proven in Corollary 1, followed by Lemma 2 and Corollary 2, where the norms for measuring the size of the data are weakened to serve the purpose of devising a finite element method in Section 3.
First we prove an auxiliary logarithmic convexity inequality, which is a more explicit version of [LRL12, Lemma 5.2].
Lemma 1. Suppose thata,b,c >= 0a, b, c \geq 0andp,q > 0p, q>0satisfyc <= bc \leq bandc <= e^(p lambda)a+e^(-q lambda)bc \leq e^{p \lambda} a+e^{-q \lambda} bfor alllambda > lambda_(0) >= 0\lambda>\lambda_{0} \geq 0. Then there areC > 0C>0andkappa in(0,1)\kappa \in(0,1)(depending only onppandqq) such that
c <= Ce^(qlambda_(0))a^(kappa)b^(1-kappa)c \leq C e^{q \lambda_{0}} a^{\kappa} b^{1-\kappa}
Proof. We may assume thata,b > 0a, b>0, sincec=0c=0ifa=0a=0orb=0b=0. The minimizerlambda_(**)\lambda_{*}of the functionf(lambda)=e^(p lambda)a+e^(-q lambda)bf(\lambda)=e^{p \lambda} a+e^{-q \lambda} bis given by
That is, iflambda_(**) <= lambda_(0)\lambda_{*} \leq \lambda_{0}then
c <= C_(2)e^(qlambda_(0))a^(kappa)b^(1-kappa)c \leq C_{2} e^{q \lambda_{0}} a^{\kappa} b^{1-\kappa}
whereC_(2)=r^(-q//(p+q))C_{2}=r^{-q /(p+q)}. Ase^(qlambda_(0)) >= 1e^{q \lambda_{0}} \geq 1andC_(1) > C_(2)C_{1}>C_{2}, the claim follows by takingC=C_(1)C=C_{1}. The following Carleman inequality is well-known, see e.g. [LRL12]. For the convenience of the reader we have included an elementary proof in Appendix A.
Proposition 1. Letrho inC^(3)(Omega)\rho \in C^{3}(\Omega)andK sub OmegaK \subset \Omegabe a compact set that does not contain critical points ofrho\rho. Letalpha,tau > 0\alpha, \tau>0andphi=e^(alpha rho)\phi=e^{\alpha \rho}. Letw inC_(0)^(2)(K)w \in C_{0}^{2}(K)andv=e^(tau phi)wv=e^{\tau \phi} w. Then there isC > 0C>0such that
int_(K)e^(2tau phi)(tau^(3)w^(2)+tau|grad w|^(2))dx <= Cint_(K)e^(2tau phi)|Delta w|^(2)dx\int_{K} e^{2 \tau \phi}\left(\tau^{3} w^{2}+\tau|\nabla w|^{2}\right) \mathrm{d} x \leq C \int_{K} e^{2 \tau \phi}|\Delta w|^{2} \mathrm{~d} x
foralpha\alphalarge enough andtau >= tau_(0)\tau \geq \tau_{0}, wheretau_(0) > 1\tau_{0}>1depends only onalpha\alphaandrho\rho. Using the above Carleman estimate we prove a three-ball inequality that is explicit with respect tomu\muandbeta\beta, i.e. the constants in the inequality are independent of the Péclet number. The corresponding inequality with constant depending implicitly on the Péclet number is proven for instance in [MV12]. We denote byB(x,r)B(x, r)the open ball of radiusrrcentred atxx, and byd(x,del Omega)d(x, \partial \Omega)the distance fromxxto the boundary ofOmega\Omega.
Corollary 1. Letx_(0)in Omegax_{0} \in \Omegaand0 < r_(1) < r_(2) < d(x_(0),del Omega)0<r_{1}<r_{2}<d\left(x_{0}, \partial \Omega\right). DefineB_(j)=B(x_(0),r_(j)),j=1,2B_{j}=B\left(x_{0}, r_{j}\right), j=1,2. Then there areC > 0C>0andkappa in(0,1)\kappa \in(0,1)such that formu > 0,beta in[L^(oo)(Omega)]^(n)\mu>0, \beta \in\left[L^{\infty}(\Omega)\right]^{n}andu inH^(2)(Omega)u \in H^{2}(\Omega)it holds that
||u||_(H^(1)(B_(2))) <= Ce^(C tilde(P)e^(2))(||u||_(H^(1)(B_(1)))+(1)/(mu)||Lu||_(L^(2)(Omega)))^(kappa)||u||_(H^(1)(Omega))^(1-kappa)\|u\|_{H^{1}\left(B_{2}\right)} \leq C e^{C \tilde{P} e^{2}}\left(\|u\|_{H^{1}\left(B_{1}\right)}+\frac{1}{\mu}\|\mathcal{L} u\|_{L^{2}(\Omega)}\right)^{\kappa}\|u\|_{H^{1}(\Omega)}^{1-\kappa}
wheretilde(P)e=1+|beta|//mu\tilde{P} e=1+|\beta| / \muand|beta|=||beta||_([L^(oo)(Omega)]^(n))|\beta|=\|\beta\|_{\left[L^{\infty}(\Omega)\right]^{n}}. Proof. Due to the density ofC^(2)(Omega)C^{2}(\Omega)inH^(2)(Omega)H^{2}(\Omega), it is enough to consideru inC^(2)(Omega)u \in C^{2}(\Omega). Let now0 < r_(0) < r_(1)0<r_{0}<r_{1}andr_(2) < r_(3) < r_(4) < d(x_(0),del Omega)r_{2}<r_{3}<r_{4}<d\left(x_{0}, \partial \Omega\right). We choose non-positiverho inC^(oo)(Omega)\rho \in C^{\infty}(\Omega)such thatrho(x)=-d(x,x_(0))\rho(x)=-d\left(x, x_{0}\right)outsideB_(0)B_{0}. Since|grad rho|=1|\nabla \rho|=1outsideB_(0),rhoB_{0}, \rhodoes not have critical points inB_(4)\\B_(0)B_{4} \backslash B_{0}. Letchi inC_(0)^(oo)(B_(4)\\B_(0))\chi \in C_{0}^{\infty}\left(B_{4} \backslash B_{0}\right)satisfychi=1\chi=1inB_(3)\\B_(1)B_{3} \backslash B_{1}, and setw=chi uw=\chi u. We apply Proposition 1 withK= bar(B_(4))\\B_(0)K=\overline{B_{4}} \backslash B_{0}to get
forphi=e^(alpha rho)\phi=e^{\alpha \rho}, with large enoughalpha > 0\alpha>0, andtau >= tau_(0)\tau \geq \tau_{0}(wheretau_(0) > 1\tau_{0}>1depends only onalpha\alphaandrho\rho). We bound from above the right-hand side by a constant times
fortau >= tau_(0)+2|beta|^(2)//mu^(2)\tau \geq \tau_{0}+2|\beta|^{2} / \mu^{2}. We divide bymu^(2)\mu^{2}and conclude by Lemma 1 withp=1-Phi(r_(2)) > 0p=1-\Phi\left(r_{2}\right)>0andq=Phi(r_(2))-Phi(r_(3)) > 0q=\Phi\left(r_{2}\right)-\Phi\left(r_{3}\right)>0, followed by absorbing thetilde(Pe)=1+|beta|//mu\tilde{P e}=1+|\beta| / \mufactor into the exponential factore^(C tilde(P)e^(2))e^{C \tilde{P} e^{2}}.
We now shift down the Sobolev indices in Corollary 1 by making a similar argument to that in Section 4 of [DSFKSU09] or Section 2.2 of [BNO19b], based on semiclassical pseudodifferential calculus.
Lemma 2. Letx_(0)in Omegax_{0} \in \Omegaand0 < r_(1) < r_(2) < d(x_(0),del Omega)0<r_{1}<r_{2}<d\left(x_{0}, \partial \Omega\right). DefineB_(j)=B(x_(0),r_(j)),j=1,2B_{j}=B\left(x_{0}, r_{j}\right), j=1,2. Then there areC > 0C>0andkappa in(0,1)\kappa \in(0,1)such that formu > 0,beta in[L^(oo)(Omega)]^(n)\mu>0, \beta \in\left[L^{\infty}(\Omega)\right]^{n}andu inH^(2)(Omega)u \in H^{2}(\Omega)it holds that
||u||_(L^(2)(B_(2))) <= Ce^(C tilde(P)e^(2))(||u||_(L^(2)(B_(1)))+(1)/(mu)||Lu||_(H^(-1)(Omega)))^(kappa)||u||_(L^(2)(Omega))^(1-kappa),\|u\|_{L^{2}\left(B_{2}\right)} \leq C e^{C \tilde{P} e^{2}}\left(\|u\|_{L^{2}\left(B_{1}\right)}+\frac{1}{\mu}\|\mathcal{L} u\|_{H^{-1}(\Omega)}\right)^{\kappa}\|u\|_{L^{2}(\Omega)}^{1-\kappa},
wheretilde(Pe)=1+|beta|//mu\tilde{P e}=1+|\beta| / \muand|beta|=||beta||_([L^(oo)(Omega)]^(n)^(n))|\beta|=\|\beta\|_{{\left[L^{\infty}(\Omega)\right]^{n}}^{n}}. Proof. Letℏ > 0\hbar>0be the semiclassical parameter that satisfiesℏ=1//tau\hbar=1 / \tau, wheretau\tauis the parameter previously introduced in Proposition 1. We will make use of the theory of semiclassical pseudodifferential operators, which we briefly recall in Appendix B for the convenience of the reader. In particular we will use semiclassical Sobolev spaces with norms given by
where the scale of the semiclassical Bessel potentials is defined by
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}
We will also use the following commutator and pseudolocal estimates, see Appendix B. Suppose thateta,vartheta inC_(0)^(oo)(R^(n))\eta, \vartheta \in C_{0}^{\infty}\left(\mathbb{R}^{n}\right)and thateta=1\eta=1nearsupp(vartheta)\operatorname{supp}(\vartheta), and letA_(psi),B_(psi)A_{\psi}, B_{\psi}be two semiclassical pseudodifferential operators of orderss,ms, m, respectively. Then for allp,q,N inRp, q, N \in \mathbb{R}, there isC > 0C>0,
We now combine this estimate with the technique used to prove Corollary 1. Consideru inC^(oo)(R^(n))u \in C^{\infty}\left(\mathbb{R}^{n}\right)and setw=e^(phi//ℏ)uw=e^{\phi / \hbar} u. Takepsi inC_(0)^(oo)(Omega)\psi \in C_{0}^{\infty}(\Omega)supported inB_(1)uu(B_(5)\\ tilde(B)_(3))B_{1} \cup\left(B_{5} \backslash \tilde{B}_{3}\right)withpsi=1\psi=1in( tilde(B)_(1)\\B_(0))uu(B_(4)\\B_(3))\left(\tilde{B}_{1} \backslash B_{0}\right) \cup\left(B_{4} \backslash B_{3}\right). Recall thatchi inC_(0)^(oo)(B_(4)\\B_(0))\chi \in C_{0}^{\infty}\left(B_{4} \backslash B_{0}\right)satisfieschi=1\chi=1inB_(3)\\ tilde(B)_(1)B_{3} \backslash \tilde{B}_{1}. Using (4) to bound the commutator
where we used the norm inequality||*||_(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)}. LettingPhi(r)=e^(-alpha r)\Phi(r)=e^{-\alpha r}and using a similar argument as in the proof of Corollary 1, we find that
whenℏ\hbarsatisfies (6) and is small enough. Absorbing the negative power ofℏ\hbarin the exponential, we then use Lemma 1 and conclude by absorbing thetilde(P)e=1+|beta|//mu\tilde{P} e=1+|\beta| / \mufactor into the exponential factore^(C tilde(P)e^(2))e^{C \tilde{P} e^{2}}.
Making the additional coercivity assumptiongrad*beta <= 0\nabla \cdot \beta \leq 0, we can weaken the norms just in the right-hand side of Corollary 1 by using the stability estimate for a well-posed convection-diffusion problem with homogeneous Dirichlet boundary conditions.
Corollary 2. Letx_(0)in Omegax_{0} \in \Omegaand0 < r_(1) < r_(2) < d(x_(0),del Omega)0<r_{1}<r_{2}<d\left(x_{0}, \partial \Omega\right). DefineB_(j)=B(x_(0),r_(j)),j=1,2B_{j}=B\left(x_{0}, r_{j}\right), j=1,2. Then there areC > 0C>0andkappa in(0,1)\kappa \in(0,1)such that formu > 0,beta in[W^(1,oo)(Omega)]^(n)\mu>0, \beta \in\left[W^{1, \infty}(\Omega)\right]^{n}havingesssup_(Omega)grad*beta <= 0\operatorname{esssup}_{\Omega} \nabla \cdot \beta \leq 0, andu inH^(2)(Omega)u \in H^{2}(\Omega)it holds that
wheretilde(Pe)=1+|beta|//mu\tilde{P e}=1+|\beta| / \muand|beta|=||beta||_([^(L^(oo)(Omega)]^(n)):})|\beta|=\|\beta\|_{\left[^{\left.L^{\infty}(\Omega)\right]^{n}}\right.}. Proof. Let the ballsB_(0),B_(3)sub OmegaB_{0}, B_{3} \subset \Omegasuch thatB_(j)subB_(j+1)B_{j} \subset B_{j+1}, forj=0,2j=0,2. Consider the wellposed problem
Lw=Lu" in "B_(3),quad w=0" on "delB_(3).\mathcal{L} w=\mathcal{L} u \text { in } B_{3}, \quad w=0 \text { on } \partial B_{3} .
Sinceesss u p_(Omega)grad*beta <= 0\operatorname{ess} \sup _{\Omega} \nabla \cdot \beta \leq 0, as a consequence of the divergence theorem we have
||w||_(H^(1)(B_(3))) <= C(1)/(mu)||Lu||_(H^(-1)(B_(3)))\|w\|_{H^{1}\left(B_{3}\right)} \leq C \frac{1}{\mu}\|\mathcal{L} u\|_{H^{-1}\left(B_{3}\right)}
Takingv=u-wv=u-w, we haveLv=0\mathcal{L} v=0inB_(3)B_{3}. The stability estimate in Corollary 1 used forB_(0),B_(2),B_(3)B_{0}, B_{2}, B_{3}reads as
||v||_(H^(1)(B_(2))) <= Ce^(C tilde(P)e^(2))||v||_(H^(1)(B_(0)))^(kappa)||v||_(H^(1)(B_(3)))^(1-kappa),\|v\|_{H^{1}\left(B_{2}\right)} \leq C e^{C \tilde{P} e^{2}}\|v\|_{H^{1}\left(B_{0}\right)}^{\kappa}\|v\|_{H^{1}\left(B_{3}\right)}^{1-\kappa},
Now we choose a cutoff functionchi inC_(0)^(oo)(B_(1))\chi \in C_{0}^{\infty}\left(B_{1}\right)such thatchi=1\chi=1inB_(0)B_{0}. Thenchi u\chi usatisfies
L(chi u)=chiLu+[L,chi]u,quad chi u=0" on "delB_(1),\mathcal{L}(\chi u)=\chi \mathcal{L} u+[\mathcal{L}, \chi] u, \quad \chi u=0 \text { on } \partial B_{1},
The same argument forB_(3)sub OmegaB_{3} \subset \Omegagives
||u||_(H^(1)(B_(3))) <= C(1)/(mu)((mu+|beta|)||u||_(L^(2)(Omega))+||Lu||_(H^(-1)(Omega)))\|u\|_{H^{1}\left(B_{3}\right)} \leq C \frac{1}{\mu}\left((\mu+|\beta|)\|u\|_{L^{2}(\Omega)}+\|\mathcal{L} u\|_{H^{-1}(\Omega)}\right)
thus leading to the conclusion after absorbing thetilde(P)e=1+|beta|//mu\tilde{P} e=1+|\beta| / \mufactor into the exponential factore^(C tilde(P)e^(2))e^{C \tilde{P} e^{2}}.
Remark 1. In the geometric setting of this section one can be more precise about the Hölder exponentkappa\kappain the conditional stability estimates. For this we recall some known results for second-order elliptic equations: we refer to [ARRV09, Theorem 2.1] for the Laplace equation, and for the case including lower-order terms to [MV12, Theorem 3]. Letuube a homogeneous solution of (1) withf-=0f \equiv 0. For a constantC_("st ")C_{\text {st }}depending implicitly on the coefficientsmu\muandbeta\beta, the following three-ball inequality holds
whereB_(j)B_{j}are concentric balls inOmega\Omegawith increasing radiir_(j)r_{j}. The constantC_(st)C_{s t}does not depend on the radiir_(1),r_(2)r_{1}, r_{2}, but it does depend onr_(3)r_{3}. The exponentkappa in(0,1)\kappa \in(0,1)is given by
whereC_(3) > 0C_{3}>0is a constant depending onr_(3)r_{3}.
3. Finite element method
LetV_(h)V_{h}denote the space of piecewise affine finite element functions defined on a conforming computational meshT_(h)={K}.T_(h)\mathcal{T}_{h}=\{K\} . \mathcal{T}_{h}consists of shape regular triangular elementsKKwith diameterh_(K)h_{K}and is quasi-uniform. We define the global mesh size byh=max_(K inT_(h))h_(K)h=\max _{K \in \mathcal{T}_{h}} h_{K}. The interior faces of the triangulation will be denoted byF_(i)\mathcal{F}_{i}, the jump of a quantity across a faceFFby[[*]]_(F)\llbracket \cdot \rrbracket_{F}, and the outward unit normal bynn.
Letbeta in[W^(1,oo)(Omega)]^(n)\beta \in\left[W^{1, \infty}(\Omega)\right]^{n}and adopt the shorthand notation|beta|:=||beta||_([L^(oo)(Omega)]^(n))|\beta|:=\|\beta\|_{\left[L^{\infty}(\Omega)\right]^{n}}. As already stated in Section 1, we consider the diffusion-dominated regime given by the low Péclet number
{:(11)Pe(h):=(|beta|h)/(mu) < 1.:}\begin{equation*} P e(h):=\frac{|\beta| h}{\mu}<1 . \tag{11} \end{equation*}
We will denote byCCa generic positive constant independent of the mesh size and the Péclet number. Letpi_(h):L^(2)(Omega)|->V_(h)\pi_{h}: L^{2}(\Omega) \mapsto V_{h}denote the standardL^(2)L^{2}-projection onV_(h)V_{h}, which fork=1,2k=1,2andm=0,k-1m=0, k-1satisfies
{:[||pi_(h)u||_(H^(m)(Omega)) <= C||u||_(H^(m)(Omega))","quad u inH^(m)(Omega)],[||u-pi_(h)u||_(H^(m)(Omega)) <= Ch^(k-m)||u||_(H^(k)(Omega))","quad u inH^(k)(Omega)]:}\begin{aligned} \left\|\pi_{h} u\right\|_{H^{m}(\Omega)} & \leq C\|u\|_{H^{m}(\Omega)}, \quad u \in H^{m}(\Omega) \\ \left\|u-\pi_{h} u\right\|_{H^{m}(\Omega)} & \leq C h^{k-m}\|u\|_{H^{k}(\Omega)}, \quad u \in H^{k}(\Omega) \end{aligned}
We introduce the standard inner products with the induced norms
The termss_(Omega)s_{\Omega}ands_(**)s_{*}are stabilizing terms, while the terms_(omega)s_{\omega}is aimed for data assimilation. After scaling with the coefficients in the above forms, Lemma 2 in [BHL18] writes as
{:(13)s_(Omega)(pi_(h)u,pi_(h)u) <= C gamma(mu+|beta|h)h^(2)|u|_(H^(2)(Omega))^(2)","quad AA u inH^(2)(Omega).:}\begin{equation*} s_{\Omega}\left(\pi_{h} u, \pi_{h} u\right) \leq C \gamma(\mu+|\beta| h) h^{2}|u|_{H^{2}(\Omega)}^{2}, \quad \forall u \in H^{2}(\Omega) . \tag{13} \end{equation*}
The parametersgamma\gammaandgamma_(**)\gamma_{*}ins_(Omega)s_{\Omega}ands_(**)s_{*}, respectively, are fixed at the implementation level and, to alleviate notation, our analysis covers the choicegamma=1=gamma_(**)\gamma=1=\gamma_{*}.
We can then use the general framework in [Bur13] to write the finite element method for unique continuation subject to (1) as follows. Consider a discrete Lagrange multiplierz_(h)inV_(h)z_{h} \in V_{h}and aim to find the saddle points of the functional
where we recall thattilde(U)_(omega)=u|_(omega)+delta\tilde{U}_{\omega}=\left.u\right|_{\omega}+\deltaandu inH^(2)(Omega)u \in H^{2}(\Omega)is a solution to (1). The Euler-Lagrange equations forL_(h)L_{h}lead to the following discrete problem: find(u_(h),z_(h))in[V_(h)]^(2)\left(u_{h}, z_{h}\right) \in\left[V_{h}\right]^{2}such that
We observe that by the ill-posed character of the problem, only the stabilization operatorss_(Omega)s_{\Omega}ands_(**)s_{*}provide some stability to the discrete system, and the corresponding system matrix is expected to be ill-conditioned. To quantify this effect we first prove an upper bound on the condition number.
Proposition 2. The finite element formulation (14) has a unique solution(u_(h),z_(h))in[V_(h)]^(2)\left(u_{h}, z_{h}\right) \in\left[V_{h}\right]^{2}and the Euclidean condition numberK_(2)\mathcal{K}_{2}of the system matrix satisfies
K_(2) <= Ch^(-4).\mathcal{K}_{2} \leq C h^{-4} .
Proof. We write (14) as the linear systemA[(u_(h),z_(h)),(v_(h),w_(h))]=(f,w_(h))_(Omega)+s_(omega)( tilde(U)_(omega),v_(h))A\left[\left(u_{h}, z_{h}\right),\left(v_{h}, w_{h}\right)\right]=\left(f, w_{h}\right)_{\Omega}+s_{\omega}\left(\tilde{U}_{\omega}, v_{h}\right), for all(v_(h),w_(h))in[V_(h)]^(2)\left(v_{h}, w_{h}\right) \in\left[V_{h}\right]^{2}, where
SinceA[(u_(h),z_(h)),(u_(h),-z_(h))]=s(u_(h),u_(h))+s_(**)(z_(h),z_(h))A\left[\left(u_{h}, z_{h}\right),\left(u_{h},-z_{h}\right)\right]=s\left(u_{h}, u_{h}\right)+s_{*}\left(z_{h}, z_{h}\right), using (12) the following inf-sup condition holds
Psi_(h):=i n f_((u_(h),z_(h))in[V_(h)]^(2))s u p_((v_(h),w_(h))in[V_(h)]^(2))(A[(u_(h),z_(h)),(v_(h),w_(h))])/(||(u_(h),z_(h))||_(L^(2)(Omega))||(v_(h),w_(h))||_(L^(2)(Omega))) >= C mu(1+Pe(h))h^(2).\Psi_{h}:=\inf _{\left(u_{h}, z_{h}\right) \in\left[V_{h}\right]^{2}} \sup _{\left(v_{h}, w_{h}\right) \in\left[V_{h}\right]^{2}} \frac{A\left[\left(u_{h}, z_{h}\right),\left(v_{h}, w_{h}\right)\right]}{\left\|\left(u_{h}, z_{h}\right)\right\|_{L^{2}(\Omega)}\left\|\left(v_{h}, w_{h}\right)\right\|_{L^{2}(\Omega)}} \geq C \mu(1+P e(h)) h^{2} .
This provides the existence of a unique solution for the linear system. We use [EG06, Theorem 3.1] to estimate the condition number by
{:(15)K_(2) <= C(Υ_(h))/(Psi_(h))",":}\begin{equation*} \mathcal{K}_{2} \leq C \frac{\Upsilon_{h}}{\Psi_{h}}, \tag{15} \end{equation*}
where
Υ_(h):=s u p_((u_(h),z_(h))in[V_(h)]^(2))s u p_((v_(h),w_(h))in[V_(h)]^(2))(A[(u_(h),z_(h)),(v_(h),w_(h))])/(||(u_(h),z_(h))||_(L^(2)(Omega))||(v_(h),w_(h))||_(L^(2)(Omega))).\Upsilon_{h}:=\sup _{\left(u_{h}, z_{h}\right) \in\left[V_{h}\right]^{2}} \sup _{\left(v_{h}, w_{h}\right) \in\left[V_{h}\right]^{2}} \frac{A\left[\left(u_{h}, z_{h}\right),\left(v_{h}, w_{h}\right)\right]}{\left\|\left(u_{h}, z_{h}\right)\right\|_{L^{2}(\Omega)}\left\|\left(v_{h}, w_{h}\right)\right\|_{L^{2}(\Omega)}} .
We recall the following discrete inverse inequality, see for instance [EG04, Lemma 1.138],
Using the Cauchy-Schwarz inequality together with (18) and (16) we get
{:[s_(Omega)(u_(h),v_(h))=gamma mu(1+Pe(h))sum_(F inF_(i))int_(F)h[[gradu_(h)*n]]_(F)[[gradv_(h)*n]]_(F)ds],[ <= C mu(1+Pe(h))h^(-2)||u_(h)||_(L^(2)(Omega))||v_(h)||_(L^(2)(Omega))","]:}\begin{aligned} s_{\Omega}\left(u_{h}, v_{h}\right) & =\gamma \mu(1+P e(h)) \sum_{F \in \mathcal{F}_{i}} \int_{F} h \llbracket \nabla u_{h} \cdot n \rrbracket_{F} \llbracket \nabla v_{h} \cdot n \rrbracket_{F} \mathrm{~d} s \\ & \leq C \mu(1+P e(h)) h^{-2}\left\|u_{h}\right\|_{L^{2}(\Omega)}\left\|v_{h}\right\|_{L^{2}(\Omega)}, \end{aligned}
hence
s(u_(h),v_(h)) <= C mu(1+Pe(h))h^(-2)||u_(h)||_(L^(2)(Omega))||v_(h)||_(L^(2)(Omega)).s\left(u_{h}, v_{h}\right) \leq C \mu(1+P e(h)) h^{-2}\left\|u_{h}\right\|_{L^{2}(\Omega)}\left\|v_{h}\right\|_{L^{2}(\Omega)} .
Combining this with the Cauchy-Schwarz inequality and the inequalities (16) and (17), we obtain
-s_(**)(z_(h),w_(h)) <= C mu(1+Pe(h))h^(-2)||z_(h)||_(L^(2)(Omega))||w_(h)||_(L^(2)(Omega)).-s_{*}\left(z_{h}, w_{h}\right) \leq C \mu(1+P e(h)) h^{-2}\left\|z_{h}\right\|_{L^{2}(\Omega)}\left\|w_{h}\right\|_{L^{2}(\Omega)} .
Again due to the Cauchy-Schwarz inequality, and trace and inverse inequalities, we have
{:[a_(h)(u_(h),w_(h))=(beta*gradu_(h),w_(h))_(Omega)+musum_(F inF_(i))int_(F)h[[gradu_(h)*n]]_(F)w_(h)ds],[ <= C mu(1+Pe(h))h^(-2)||u_(h)||_(L^(2)(Omega))||w_(h)||_(L^(2)(Omega))]:}\begin{aligned} a_{h}\left(u_{h}, w_{h}\right) & =\left(\beta \cdot \nabla u_{h}, w_{h}\right)_{\Omega}+\mu \sum_{F \in \mathcal{F}_{i}} \int_{F} h \llbracket \nabla u_{h} \cdot n \rrbracket_{F} w_{h} \mathrm{~d} s \\ & \leq C \mu(1+P e(h)) h^{-2}\left\|u_{h}\right\|_{L^{2}(\Omega)}\left\|w_{h}\right\|_{L^{2}(\Omega)} \end{aligned}
Collecting the above estimates we haveΥ_(h) <= C mu(1+Pe(h))h^(-2)\Upsilon_{h} \leq C \mu(1+P e(h)) h^{-2}, and we conclude by (15). 3.1. Error estimates for the weakly consistent regularization. The error analysis proceeds in two main steps: (i) First we prove that the stabilizing terms and the data fitting term must vanish at an optimal rate for smooth solutions, with constant independent of the physical stability (Proposition 3). (ii) Then we show that the residual of the PDE is bounded by the stabilizing terms and the data fitting term. Using this result together with the first step and the continuous stability estimates in Section 2, we proveL^(2)L^{2}- andH^(1)H^{1}-convergence results (Theorems 1 and 2). To quantify stabilization and data fitting for(v_(h),w_(h))in[V_(h)]^(2)\left(v_{h}, w_{h}\right) \in\left[V_{h}\right]^{2}we introduce the norm
{:[||(u-pi_(h)u,0)||_(s)^(2)=s(u-pi_(h)u,u-pi_(h)u)=s_(Omega)(pi_(h)u,pi_(h)u)+s_(omega)(u-pi_(h)u,u-pi_(h)u)],[ <= C(muh^(2)+|beta|h^(3))|u|_(H^(2)(Omega))^(2)]:}\begin{aligned} \left\|\left(u-\pi_{h} u, 0\right)\right\|_{s}^{2} & =s\left(u-\pi_{h} u, u-\pi_{h} u\right)=s_{\Omega}\left(\pi_{h} u, \pi_{h} u\right)+s_{\omega}\left(u-\pi_{h} u, u-\pi_{h} u\right) \\ & \leq C\left(\mu h^{2}+|\beta| h^{3}\right)|u|_{H^{2}(\Omega)}^{2} \end{aligned}
where we used thats_(Omega)(u,v_(h))=0s_{\Omega}\left(u, v_{h}\right)=0, sinceu inH^(2)(Omega)u \in H^{2}(\Omega). Hence it follows that foru inH^(2)(Omega)u \in H^{2}(\Omega)
Observe that, whenPe(h) < 1P e(h)<1, the first term dominates and the estimate isO(h)O(h), whereas whenPe(h) > 1P e(h)>1the bound isO(h^((3)/(2)))O\left(h^{\frac{3}{2}}\right). We note in passing that the same estimates hold for the nodal interpolant.
Lemma 3 (Consistency). Letu inH^(2)(Omega)u \in H^{2}(\Omega)be a solution to (1) and ({:u_(h),z_(h))in[V_(h)]^(2)\left.u_{h}, z_{h}\right) \in\left[V_{h}\right]^{2}the solution to (14), then
for all(v_(h),w_(h))in[V_(h)]^(2)\left(v_{h}, w_{h}\right) \in\left[V_{h}\right]^{2}. Proof. The first claim follows from the definition ofa_(h)a_{h}, since
Proof. Writing out the terms ofa_(h)a_{h}and integrating by parts in the advective term leads to a_(h)(v,w_(h))=-(v,beta*gradw_(h))_(Omega)-(v grad*beta,w_(h))_(Omega)+(:v beta*n,w_(h):)_(del Omega)+(mu grad v,gradw_(h))_(Omega)-(:mu grad v*n,w_(h):)_(del Omega)a_{h}\left(v, w_{h}\right)=-\left(v, \beta \cdot \nabla w_{h}\right)_{\Omega}-\left(v \nabla \cdot \beta, w_{h}\right)_{\Omega}+\left\langle v \beta \cdot n, w_{h}\right\rangle_{\partial \Omega}+\left(\mu \nabla v, \nabla w_{h}\right)_{\Omega}-\left\langle\mu \nabla v \cdot n, w_{h}\right\rangle_{\partial \Omega}. Using the Cauchy-Schwarz inequality and the trace inequality (17) forvv, we see that
(:v beta*n,w_(h):)_(del Omega)+(mu grad v,gradw_(h))_(Omega)-(:mu grad v*n,w_(h):)_(del Omega) <= C||v||_(♯)||(0,w_(h))||_(s).\left\langle v \beta \cdot n, w_{h}\right\rangle_{\partial \Omega}+\left(\mu \nabla v, \nabla w_{h}\right)_{\Omega}-\left\langle\mu \nabla v \cdot n, w_{h}\right\rangle_{\partial \Omega} \leq C\|v\|_{\sharp}\left\|\left(0, w_{h}\right)\right\|_{s} .
By the Cauchy-Schwarz inequality and a discrete Poincaré inequality forw_(h)w_{h}, see e.g. [Bre03], we bound
-(v grad*beta,w_(h))_(Omega) <= C|beta|_(1,oo)||v||_(Omega)||w_(h)||_(Omega) <= C(|beta|_(1,oo))/(|beta|)Pe(h)^((1)/(2))||v||_(♯)||(0,w_(h))||_(s)-\left(v \nabla \cdot \beta, w_{h}\right)_{\Omega} \leq C|\beta|_{1, \infty}\|v\|_{\Omega}\left\|w_{h}\right\|_{\Omega} \leq C \frac{|\beta|_{1, \infty}}{|\beta|} P e(h)^{\frac{1}{2}}\|v\|_{\sharp}\left\|\left(0, w_{h}\right)\right\|_{s}
Under the assumption|beta|_(1,oo) <= C|beta||\beta|_{1, \infty} \leq C|\beta|, we get
-(v grad*beta,w_(h))_(Omega) <= CPe(h)^((1)/(2))||v||_(♯)||(0,w_(h))||_(s).-\left(v \nabla \cdot \beta, w_{h}\right)_{\Omega} \leq C P e(h)^{\frac{1}{2}}\|v\|_{\sharp}\left\|\left(0, w_{h}\right)\right\|_{s} .
We bound the remaining term by
-(v,beta*gradw_(h))_(Omega) <= |beta|^((1)/(2))h^((1)/(2))||v||_(♯)||gradw_(h)||_(Omega) <= CPe(h)^((1)/(2))||v||_(♯)||(0,w_(h))||_(s).-\left(v, \beta \cdot \nabla w_{h}\right)_{\Omega} \leq|\beta|^{\frac{1}{2}} h^{\frac{1}{2}}\|v\|_{\sharp}\left\|\nabla w_{h}\right\|_{\Omega} \leq C P e(h)^{\frac{1}{2}}\|v\|_{\sharp}\left\|\left(0, w_{h}\right)\right\|_{s} .
Finally, exploiting the low Péclet regimePe(h) < 1P e(h)<1, we obtain the conclusion. Proposition 3 (Convergence of regularization). Assume the low Péclet regime (11) and that|beta|_(1,oo) <= C|beta||\beta|_{1, \infty} \leq C|\beta|. Letu inH^(2)(Omega)u \in H^{2}(\Omega)be a solution to (1) and ({:u_(h),z_(h))in[V_(h)]^(2)\left.u_{h}, z_{h}\right) \in\left[V_{h}\right]^{2}the solution to (14), then
and the claim follows by applying the approximation (19). Lemma 5 (Covergence of the convective term). Assume the low Péclet regime (11) and that|beta|_(1,oo) <= C|beta||\beta|_{1, \infty} \leq C|\beta|. Letu inH^(2)(Omega)u \in H^{2}(\Omega)be a solution to (1),(u_(h),z_(h))in[V_(h)]^(2)\left(u_{h}, z_{h}\right) \in\left[V_{h}\right]^{2}the solution to (14) andw inH_(0)^(1)(Omega)w \in H_{0}^{1}(\Omega), then
Proof. Denote bybeta_(h)in[V_(h)]^(n)\beta_{h} \in\left[V_{h}\right]^{n}a piecewise linear approximation ofbeta\betathat isL^(oo)L^{\infty}-stable and for which
||beta-beta_(h)||_(0,oo) <= Ch|beta|_(1,oo)\left\|\beta-\beta_{h}\right\|_{0, \infty} \leq C h|\beta|_{1, \infty}
and recall the approximation estimate in [Bur05, Theorem 2.2]
We now combine these results with the conditional stability estimates from Section 2 to obtain error bounds for the discrete solution. For this purpose, we consider an open bounded setB sub OmegaB \subset \Omegathat contains the data regionomega\omegasuch thatB\\omegaB \backslash \omegadoes not touch the boundary ofOmega\Omega. Then the estimates in Lemma 2 and Corollary 2 hold true by a covering argument, see e.g. [MV12], and we obtain local error estimates inBB. For global unique continuation fromomega\omegato the entireOmega\Omega, however, the stability deteriorates and it is of a different nature: the modulus of continuity for the given data is not of Hölder type|*|^(kappa)|\cdot|^{\kappa}any more, but of a logarithmic kind|log(*)|^(-kappa)|\log (\cdot)|^{-\kappa}.
Theorem 1 (L^(2)L^{2}-error estimate). Assume the low Péclet regime (11) and that|beta|_(1,oo) <= C|beta||\beta|_{1, \infty} \leq C|\beta|. Consideromega sub B sub Omega\omega \subset B \subset \Omegasuch thatbar(B\\omega)sub Omega\overline{B \backslash \omega} \subset \Omega. Letu inH^(2)(Omega)u \in H^{2}(\Omega)be a solution to (1) and(u_(h),z_(h))in[V_(h)]^(2)\left(u_{h}, z_{h}\right) \in\left[V_{h}\right]^{2}the solution to (14), then there iskappa in(0,1)\kappa \in(0,1)such that
||u-u_(h)||_(L^(2)(B)) <= Ch^(kappa)e^(C tilde(P)e^(2))(||u||_(H^(2)(Omega))+h^(-1)||delta||_(omega)),\left\|u-u_{h}\right\|_{L^{2}(B)} \leq C h^{\kappa} e^{C \tilde{P} e^{2}}\left(\|u\|_{H^{2}(\Omega)}+h^{-1}\|\delta\|_{\omega}\right),
wheretilde(Pe)=1+|beta|//mu\tilde{P e}=1+|\beta| / \mu. Proof. Let us consider the residual defined by(:r,w:)=a_(h)(u_(h),w)-(:f,w:)\langle r, w\rangle=a_{h}\left(u_{h}, w\right)-\langle f, w\rangle, forw inH_(0)^(1)(Omega)w \in H_{0}^{1}(\Omega). Using (14) we obtain
We split the first term in the right-hand side into convective and non-convective terms, and for the latter we integrate by parts on each elementKKand use Cauchy-Schwarz followed by the trace inequality (17) to get
{:[(mu gradu_(h),grad(w-pi_(h)w))_(Omega)-(:mu gradu_(h)*n,w-pi_(h)w:)_(del Omega)=sum_(F inF_(i))int_(F)mu[[gradu_(h)*n]]_(F)(w-pi_(h)w)ds],[ <= C mu(mu+|beta|h)^(-(1)/(2))s_(Omega)(u_(h),u_(h))^((1)/(2))(h^(-1)||w-pi_(h)w||_(L^(2)(Omega))+||w-pi_(h)w||_(H^(1)(Omega)))]:}\begin{aligned} & \left(\mu \nabla u_{h}, \nabla\left(w-\pi_{h} w\right)\right)_{\Omega}-\left\langle\mu \nabla u_{h} \cdot n, w-\pi_{h} w\right\rangle_{\partial \Omega}=\sum_{F \in \mathcal{F}_{i}} \int_{F} \mu \llbracket \nabla u_{h} \cdot n \rrbracket_{F}\left(w-\pi_{h} w\right) \mathrm{d} s \\ & \leq C \mu(\mu+|\beta| h)^{-\frac{1}{2}} s_{\Omega}\left(u_{h}, u_{h}\right)^{\frac{1}{2}}\left(h^{-1}\left\|w-\pi_{h} w\right\|_{L^{2}(\Omega)}+\left\|w-\pi_{h} w\right\|_{H^{1}(\Omega)}\right) \end{aligned}
Using (21) and interpolation we obtain
(mu gradu_(h),grad(w-pi_(h)w))_(Omega)-(:mu gradu_(h)*n,w-pi_(h)w:)_(del Omega) <= C mu(h|u|_(H^(2)(Omega))+||delta||_(omega))||w||_(H^(1)(Omega)).\left(\mu \nabla u_{h}, \nabla\left(w-\pi_{h} w\right)\right)_{\Omega}-\left\langle\mu \nabla u_{h} \cdot n, w-\pi_{h} w\right\rangle_{\partial \Omega} \leq C \mu\left(h|u|_{H^{2}(\Omega)}+\|\delta\|_{\omega}\right)\|w\|_{H^{1}(\Omega)} .
We bound the convective term ina_(h)(u_(h),w-pi_(h)w)a_{h}\left(u_{h}, w-\pi_{h} w\right)by Lemma 5, hence obtaining
where for the boundary term we used thatw|_(del Omega)=0\left.w\right|_{\partial \Omega}=0together with interpolation and (17). Bounding||(0,z_(h))||_(s)\left\|\left(0, z_{h}\right)\right\|_{s}by Proposition 3, we get
where we have absorbed thetilde(P)e=1+|beta|//mu\tilde{P} e=1+|\beta| / \mufactor into the exponential factore^(C tilde(P)e^(2))e^{C \tilde{P} e^{2}}. Remark 2. Let us briefly discuss the effect of decreasing the size of the data regionomega\omegaby considering the case of balls, that isomega=B(x_(0),r_(1))\omega=B\left(x_{0}, r_{1}\right)andB=B(x_(0),r_(2))B=B\left(x_{0}, r_{2}\right), withx_(0)in Omegax_{0} \in \Omegaandr_(1) < r_(2)r_{1}<r_{2}. Notice from Remark 1 that the exponentkappa\kappais an increasing function of the radiusr_(1)r_{1}and that decreasing the size of the data regionomega\omegaimplies that the convergence rateh^(kappa)h^{\kappa}decreases as well. Bounding the radiusr_(2)r_{2}away from zero and lettingr_(1)rarr0r_{1} \rightarrow 0implies that the exponentkappa rarr0\kappa \rightarrow 0. The continuum three-ball inequality then becomes the trivial inequality||u||_(L^(2)(B)) <= ||u||_(L^(2)(Omega))\|u\|_{L^{2}(B)} \leq\|u\|_{L^{2}(\Omega)}and the method does not converge any more. Theorem 2 (H^(1)H^{1}-error estimate). Assume the low Péclet regime (11) and that|beta|_(1,oo) <= C|beta||\beta|_{1, \infty} \leq C|\beta|andesss u p_(Omega)grad*beta <= 0\operatorname{ess} \sup _{\Omega} \nabla \cdot \beta \leq 0. Consideromega sub B sub Omega\omega \subset B \subset \Omegasuch thatbar(B\\omega)sub Omega\overline{B \backslash \omega} \subset \Omega. Letu inH^(2)(Omega)u \in H^{2}(\Omega)be a solution to (1), and(u_(h),z_(h))in[V_(h)]^(2)\left(u_{h}, z_{h}\right) \in\left[V_{h}\right]^{2}the solution to (14), then there iskappa in(0,1)\kappa \in(0,1)such that
||u-u_(h)||_(H^(1)(B)) <= Ch^(kappa)e^(C tilde(Pe)^(2))(||u||_(H^(2)(Omega))+h^(-1)||delta||_(omega)),\left\|u-u_{h}\right\|_{H^{1}(B)} \leq C h^{\kappa} e^{C \tilde{P e}^{2}}\left(\|u\|_{H^{2}(\Omega)}+h^{-1}\|\delta\|_{\omega}\right),
wheretilde(Pe)=1+|beta|//mu\tilde{P e}=1+|\beta| / \mu. Proof. Lettinge_(h)=u-u_(h)e_{h}=u-u_{h}, we combine the proof of Theorem 1 with the stability estimate in Corollary 2 to obtain
We illustrate the theoretical results with some numerical examples. The implementation of the stabilized FEM (14) has been carried out in FreeFem++ [Hec12] on uniform triangulations with alternating left and right diagonals. The mesh size is taken as the inverse square root of the number of nodes. The parameters ins_(Omega)s_{\Omega}ands_(**)s_{*}are set togamma=10^(-5)\gamma=10^{-5}andgamma_(**)=1\gamma_{*}=1. We also rescale the boundary term ins_(**)s_{*}by the factor 50 , drawing on results from different numerical experiments. In this section we denotee_(h)=pi_(h)u-u_(h)e_{h}=\pi_{h} u-u_{h}.
We considerOmega\Omegato be the unit square and the exact solution with global unitL^(2)L^{2}-norm
u(x,y)=30 x(1-x)y(1-y).u(x, y)=30 x(1-x) y(1-y) .
We take the diffusion coefficientmu=1\mu=1and investigate two cases for the convection field: the coercive case of the constant field
beta_(c)=(1,0),\beta_{c}=(1,0),
and the case
beta_(nc)=100(x+y,y-x),\beta_{n c}=100(x+y, y-x),
plotted in Figure 2, for whichgrad*beta=200\nabla \cdot \beta=200and||beta||_(0,oo)=200\|\beta\|_{0, \infty}=200. This makes the (well-posed) problem strongly non-coercive with a medium high Péclet number. The latter example was also considered in [Bur13] for numerical experiments on a non-coercive convectiondiffusion equation with Cauchy data.
We consider the following domains for data assimilation, shown in Figure 1,
The condition number upper bound in Proposition 2 is illustrated for a particular case in Figure 2, where we plot the condition numberK_(2)\mathcal{K}_{2}versus the mesh sizehh, together with reference dotted lines proportional toh^(-3)h^{-3}andh^(-4)h^{-4}. For five meshes with2^(N)2^{N}elements on each side,N=3,dots,7N=3, \ldots, 7, the approximate rates forK_(2)\mathcal{K}_{2}are-3.03,-3.16,-3.2,-3.34-3.03,-3.16,-3.2,-3.34.
The results in Figure 3 for the domains (22) strongly agree with the convergence rates expected from Theorem 1 and Theorem 2 for the relative errors inBBcomputed in theL^(2)L^{2}- andH^(1)H^{1}-norms, and with the rates for||(e_(h),z_(h))||_(s)\left\|\left(e_{h}, z_{h}\right)\right\|_{s}given in Proposition 3.
The numerical approximation improves when considering the setting in (23), in which data is given both downstream and upstream, as reported in Figure 4. The convergence is almost linear and the size of the errors is considerably reduced in the non-coercive case.
The resolution increases all the more when data is given near a big part of the boundarydel Omega\partial \Omega, as for the computational domains (24) considered in Figure 5. In this configuration of the setomega\omega, for both convective fieldsbeta_(c)\beta_{c}andbeta_(nc)\beta_{n c}, theL^(2)L^{2}-errors decrease below10^(-4)10^{-4}with superlinear rates on the same meshes considered in Figure 3 and Figure 4.
Comparing the geometries in (22) and (23) we also expect to see different effects of the two convective fieldsbeta_(c)\beta_{c}andbeta_(nc)\beta_{n c}. Notice that for both geometries the horizontal magnitude ofbeta_(nc)\beta_{n c}is greater than that ofbeta_(c)\beta_{c}. In (22) the solution is continued in the
Figure 2. Left: convection fieldbeta_(nc)\beta_{n c}. Right: condition numberK_(2)\mathcal{K}_{2}for domains (22),beta=beta_(c)\beta=\beta_{c}; the dotted lines are proportional toh^(-3)h^{-3}andh^(-4)h^{-4}.
Figure 3. Convergence for domains (22). Left:beta=beta_(c)\beta=\beta_{c}. Right:beta=beta_(nc)\beta=\beta_{n c}.
crosswind direction for bothbeta_(c)\beta_{c}andbeta_(nc)\beta_{n c}, and a stronger convective field is not expected to improve the reconstruction. On the other side, in (23) information is propagated both downstream and upstream, and a stronger convective field can improve the resolution, despite the increase in the Péclet number. Indeed, we can see in Figure 3 that for the geometry in (22) the numerical approximation is better forbeta_(c)\beta_{c}than forbeta_(nc)\beta_{n c}, while Figure 4 shows better results forbeta_(nc)\beta_{n c}than forbeta_(c)\beta_{c}in the case of (23), especially for theL^(2)L^{2}-error.
To exemplify the noisy datatilde(U)_(omega)=u|_(omega)+delta\tilde{U}_{\omega}=\left.u\right|_{\omega}+\delta, we perturb the restriction ofuutoomega\omegaon every node of the mesh with uniformly distributed values in[-h^((1)/(2)),h^((1)/(2))]\left[-h^{\frac{1}{2}}, h^{\frac{1}{2}}\right], respectively[-h,h][-h, h]. Recall that by the error estimates in Section 3 the contribution of the perturbationdelta\delta
Figure 4. Convergence for domains (23). Left:beta=beta_(c)\beta=\beta_{c}. Right:beta=beta_(nc)\beta=\beta_{n c}.
Figure 5. Convergence for domains (24). Left:beta=beta_(c)\beta=\beta_{c}. Right:beta=beta_(nc)\beta=\beta_{n c}.
is bounded byh^(-1)||delta||_(omega)h^{-1}\|\delta\|_{\omega}. It can be seen in Figure 6 that the perturbations are strongly visible for anO(h^((1)/(2)))O\left(h^{\frac{1}{2}}\right)amplitude, but not for anO(h)O(h)one.
Appendix A.
Denote by(*,*),|*|(\cdot, \cdot),|\cdot|, div,grad\nablaandD^(2)D^{2}the inner product, norm, divergence, gradient and Hessian in the Euclidean setting ofOmega subR^(n)\Omega \subset \mathbb{R}^{n}. We recall the following identity [BNO19b, Lemma 1].
Lemma 6. Letℓ,w inC^(2)(Omega)\ell, w \in C^{2}(\Omega)andsigma inC^(1)(Omega)\sigma \in C^{1}(\Omega). We definev=e^(ℓ)wv=e^{\ell} wand
we conclude by integrating overKKand using the divergence theorem.
Appendix B.
We briefly recall herein the definition of semiclassical pseudodifferential operators and semiclassical Sobolev spaces. We then discuss the composition rule of two such operators, which is also called symbol calculus, and some estimates that are used in the proof of Lemma 2. This presentation is based on [Zwo12, Chapter 4] and [LRL12, Section 2], to which we refer the reader for more details.
We shall use the following standard notation. Forxi inR^(n)\xi \in \mathbb{R}^{n}we set(:xi:)=(1+|xi|^(2))^((1)/(2))\langle\xi\rangle=\left(1+|\xi|^{2}\right)^{\frac{1}{2}}, and for a multi-indexalpha=(alpha_(1),dots,alpha_(n))inN^(n)\alpha=\left(\alpha_{1}, \ldots, \alpha_{n}\right) \in \mathbb{N}^{n}let|alpha|=alpha_(1)+dotsalpha_(n),alpha!=alpha_(1)!cdotsalpha_(n)!|\alpha|=\alpha_{1}+\ldots \alpha_{n}, \alpha!=\alpha_{1}!\cdots \alpha_{n}!,xi^(alpha)=xi_(1)^(alpha_(1))cdotsxi_(n)^(alpha_(n))\xi^{\alpha}=\xi_{1}^{\alpha_{1}} \cdots \xi_{n}^{\alpha_{n}}. Let alsodel^(alpha)=del_(x_(1))^(alpha_(1))cdotsdel_(x_(n))^(alpha_(n)),D=(1)/(i)del\partial^{\alpha}=\partial_{x_{1}}^{\alpha_{1}} \cdots \partial_{x_{n}}^{\alpha_{n}}, D=\frac{1}{i} \partialandD^(alpha)=(1)/(i|^(alpha∣))del^(alpha)D^{\alpha}=\frac{1}{\left.i\right|^{\alpha \mid}} \partial^{\alpha}. The Schwartz spaceS(R^(n))\mathcal{S}\left(\mathbb{R}^{n}\right)is the set of rapidly decreasingC^(oo)C^{\infty}functions and its dualS^(')(R^(n))\mathcal{S}^{\prime}\left(\mathbb{R}^{n}\right)is the set of tempered distributions. The semiclassical parameterℏ\hbaris assumed to be small:ℏin(0,ℏ_(0))\hbar \in\left(0, \hbar_{0}\right)withℏ_(0)≪1\hbar_{0} \ll 1.
The semiclassical Fourier transform is a rescaled version of the standard Fourier transform. It is given by
F_(ℏ)varphi(xi):=int_(R^(n))e^(-(i)/(ℏ)x*xi)varphi(x)dx\mathcal{F}_{\hbar} \varphi(\xi):=\int_{\mathbb{R}^{n}} e^{-\frac{i}{\hbar} x \cdot \xi} \varphi(x) \mathrm{d} x
The following properties hold:F_(ℏ)((ℏD_(x))^(alpha)varphi)=xi^(alpha)F_(ℏ)varphi\mathcal{F}_{\hbar}\left(\left(\hbar D_{x}\right)^{\alpha} \varphi\right)=\xi^{\alpha} \mathcal{F}_{\hbar} \varphiand(ℏD_(xi))^(alpha)F_(ℏ)varphi(xi)=F_(ℏ)((-x)^(alpha)varphi)\left(\hbar D_{\xi}\right)^{\alpha} \mathcal{F}_{\hbar} \varphi(\xi)=\mathcal{F}_{\hbar}\left((-x)^{\alpha} \varphi\right). B.1. Symbol classes. Form inRm \in \mathbb{R}the symbol classS^(m)S^{m}consists of functionsa(x,xi,ℏ)inC^(oo)(R^(n)xxR^(n))a(x, \xi, \hbar) \in C^{\infty}\left(\mathbb{R}^{n} \times \mathbb{R}^{n}\right)such that for all multi-indicesalpha, tilde(alpha)inN^(n)\alpha, \tilde{\alpha} \in \mathbb{N}^{n}there exists a constantC_(alpha, tilde(alpha)) > 0C_{\alpha, \tilde{\alpha}}>0uniform inℏin(0,ℏ_(0))\hbar \in\left(0, \hbar_{0}\right)such that
Lemma (Asymptotic series). Letm inRm \in \mathbb{R}and the symbolsa_(j)inS^(m-j)a_{j} \in S^{m-j}forj=0,1,dotsj=0,1, \ldots. Then there exists a symbola inS^(m)a \in S^{m}such thata∼sum_(j=0)^(oo)ℏ^(j)a_(j)a \sim \sum_{j=0}^{\infty} \hbar^{j} a_{j}, that is for everyN inNN \in \mathbb{N},
The symbol a is unique up toℏ^(oo)S^(-oo)\hbar^{\infty} S^{-\infty}, in the sense that the difference of two such symbols is inℏ^(N)S^(-M)\hbar^{N} S^{-M}for allN,M inRN, M \in \mathbb{R}. The principal symbol ofaais given bya_(0)a_{0}. B.2. Pseudodifferential operators. Using these symbol classes we can define semiclassical pseudodifferential operators (psiDOs\psi \mathrm{DOs}). For a symbola inS^(m)a \in S^{m}we define the corresponding semiclassicalpsiDO\psi \mathrm{DO}of orderm,Op(a):S(R^(n))rarrS(R^(n))m, O p(a): \mathcal{S}\left(\mathbb{R}^{n}\right) \rightarrow \mathcal{S}\left(\mathbb{R}^{n}\right),
This is also called quantization of the symbol.Op(a)O p(a)can be extended toS^(')(R^(n))\mathcal{S}^{\prime}\left(\mathbb{R}^{n}\right)andOp(a):S^(')(R^(n))rarrS^(')(R^(n))O p(a): \mathcal{S}^{\prime}\left(\mathbb{R}^{n}\right) \rightarrow \mathcal{S}^{\prime}\left(\mathbb{R}^{n}\right)continuously. Note thatOp(a)u(x)=F_(ℏ)^(-1)(a(x,*)F_(ℏ)u(*))O p(a) u(x)=\mathcal{F}_{\hbar}^{-1}\left(a(x, \cdot) \mathcal{F}_{\hbar} u(\cdot)\right)and that the operator corresponding to the symbola(x,xi)=sum_(|alpha| <= N)a_(alpha)(x)xi^(alpha)a(x, \xi)=\sum_{|\alpha| \leq N} a_{\alpha}(x) \xi^{\alpha}isOp(a)u=sum_(|alpha| <= N)a_(alpha)(x)(ℏD)^(alpha)uO p(a) u= \sum_{|\alpha| \leq N} a_{\alpha}(x)(\hbar D)^{\alpha} u. Notice that each derivative of this operator scales withℏ\hbar.
For the present paper the most important example is the second order differential operatorA=-ℏ^(2)Delta+ℏ^(2)sum_(j=1)^(n)beta_(j)(x)del_(j)A=-\hbar^{2} \Delta+\hbar^{2} \sum_{j=1}^{n} \beta_{j}(x) \partial_{j}. Its symbol is given bya(x,xi,ℏ)=|xi|^(2)+iℏsum_(j=1)^(n)beta_(j)(x)xi_(j)a(x, \xi, \hbar)=|\xi|^{2}+i \hbar \sum_{j=1}^{n} \beta_{j}(x) \xi_{j}, and its principal symbol isa_(0)(x,xi,ℏ)=|xi|^(2)a_{0}(x, \xi, \hbar)=|\xi|^{2}. B.3. Semiclassical Sobolev spaces. Fors inRs \in \mathbb{R}the semiclassical Sobolev spacesH_(scl)^(s)(R^(n))H_{\mathrm{scl}}^{s}\left(\mathbb{R}^{n}\right)are algebraically equal to the standard Sobolev spacesH^(s)(R^(n))H^{s}\left(\mathbb{R}^{n}\right)but are endowed with different norms
where the semiclassical Bessel potential is defined byJ^(s)=Op((:xi:)^(s))J^{s}=O p\left(\langle\xi\rangle^{s}\right). Informally,
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}
For example,||u||_(H_(scl)^(1)(R^(n)))^(2)=||u||_(L^(2)(R^(n)))^(2)+||ℏgrad u||_(L^(2)(R^(n)))^(2)\|u\|_{H_{\mathrm{scl}}^{1}\left(\mathbb{R}^{n}\right)}^{2}=\|u\|_{L^{2}\left(\mathbb{R}^{n}\right)}^{2}+\|\hbar \nabla u\|_{L^{2}\left(\mathbb{R}^{n}\right)}^{2}. A semiclassicalpsiDO\psi \mathrm{DO}of ordermmis continuous fromH_(scl)^(s)(R^(n))H_{\mathrm{scl}}^{s}\left(\mathbb{R}^{n}\right)toH_(scl)^(s-m)(R^(n))H_{\mathrm{scl}}^{s-m}\left(\mathbb{R}^{n}\right). B.4. Composition. Composition of semiclassicalpsiDO\psi \mathrm{DO}s can be analysed using the following calculus.
Theorem (Symbol calculus). Leta inS^(m)a \in S^{m}andb inS^(m^('))b \in S^{m^{\prime}}. ThenOp(a)@Op(b)=Op(a#b)O p(a) \circ O p(b)=O p(a \# b)for a certaina#b inS^(m+m^('))a \# b \in S^{m+m^{\prime}}that admits the following asymptotic series
The commutator and disjoint support estimates (4) and (5) follow, respectively, from the following.
Corollary (Commutator and disjoint support). Leta inS^(m)a \in S^{m}andb inS^(m^('))b \in S^{m^{\prime}}. Then (i)a#b-b#a inℏS^(m+m^(')-1)a \# b-b \# a \in \hbar S^{m+m^{\prime}-1}. (ii) Ifsupp(a)nn supp(b)=O/\operatorname{supp}(a) \cap \operatorname{supp}(b)=\emptyset, thena#b inℏ^(oo)S^(-oo)a \# b \in \hbar^{\infty} S^{-\infty}, i.e.a#b inℏ^(N)S^(-M)a \# b \in \hbar^{N} S^{-M}for allN,M inRN, M \in \mathbb{R}.
Proof. (i) The principal symbol ofa#ba \# b, that is the first term in its asymptotic series, isaba b. The second term is(ℏ)/(i)sum_(j=1)^(n)del_(xi_(j))a(x,xi,ℏ)del_(x_(j))b(x,xi,ℏ)\frac{\hbar}{i} \sum_{j=1}^{n} \partial_{\xi_{j}} a(x, \xi, \hbar) \partial_{x_{j}} b(x, \xi, \hbar). We thus have that the principal symbol of the commutator[Op(a),Op(b)]=Op(a#b-b#a)[O p(a), O p(b)]=O p(a \# b-b \# a)is given by
(ℏ)/(i)sum_(j=1)^(n)(del_(xi_(j))adel_(x_(j))b-del_(x_(j))adel_(xi_(j))b)inℏS^(m+m^(')-1)\frac{\hbar}{i} \sum_{j=1}^{n}\left(\partial_{\xi_{j}} a \partial_{x_{j}} b-\partial_{x_{j}} a \partial_{\xi_{j}} b\right) \in \hbar S^{m+m^{\prime}-1}
(ii) Ifsupp(a)nn supp(b)=O/\operatorname{supp}(a) \cap \operatorname{supp}(b)=\emptyset, then each term in the asymptotic series ofa#ba \# bvanishes.
Acknowledgements
E.B. was supported by EPSRC grants EP/P01576X/1 and EP/P012434/1, and L.O. by EPSRC grants EP/P01593X/1 and EP/R002207/1.
References
[ARRV09]
G. Alessandrini, L. Rondi, E. Rosset, and S. Vessella. The stability for the Cauchy problem for elliptic equations. Inverse Problems, 25:123004, 2009.
[BHL18]
E. Burman, P. Hansbo, and M. G. Larson. Solving ill-posed control problems by stabilized finite element methods: an alternative to Tikhonov regularization. Inverse Problems, 34:035004, 2018.
[BNO19a]
E. Burman, M. Nechita, and L. Oksanen. A stabilized finite element method for inverse problems subject to the convection-diffusion equation. II: convection-dominated regime. in preparation, 2019.
[BNO19b]
E. Burman, M. Nechita, and L. Oksanen. Unique continuation for the Helmholtz equation using stabilized finite element methods. J. Math. Pures Appl., 129:1-24, 2019.
[BO18]
E. Burman and L. Oksanen. Data assimilation for the heat equation using stabilized finite element methods. Numer. Math., 139(3):505-528, 2018.
[Bre03]
S. C. Brenner. Poincaré-Friedrichs inequalities for piecewiseH^(1)H^{1}functions. SIAM J. Numer. Anal., 41(1):306-324, 2003.
[Bur05]
E. Burman. A unified analysis for conforming and nonconforming stabilized finite element methods using interior penalty. SIAM J. Numer. Anal., 43(5):2012-2033, 2005.
[Bur13]
E. Burman. Stabilized finite element methods for nonsymmetric, noncoercive, and ill-posed problems. Part I: elliptic equations. SIAM J. Sci. Comput., 35(6):A2752-A2780, 2013.
[Bur14]
E. Burman. Error estimates for stabilized finite element methods applied to ill-posed problems. C. R. Math. Acad. Sci. Paris, 352(7-8):655-659, 2014.
[BV07]
R. Becker and B. Vexler. Optimal control of the convection-diffusion equation using stabilized finite element methods. Numer. Math., 106(3):349-367, 2007.
[DQ05]
L. Dede' and A. Quarteroni. Optimal control and numerical adaptivity for advectiondiffusion equations. M2AN Math. Model. Numer. Anal., 39(5):1019-1040, 2005.
[DSFKSU09]
D. Dos Santos Ferreira, C. E. Kenig, M. Salo, and G. Uhlmann. Limiting Carleman weights and anisotropic inverse problems. Invent. Math., 178(1):119-171, 2009.
[EG04]
A. Ern and J.-L. Guermond. Theory and practice of finite elements, volume 159 of Applied Mathematical Sciences. Springer-Verlag, New York, 2004.
[EG06]
A. Ern and J.-L. Guermond. Evaluation of the condition number in linear systems arising in finite element approximations. M2AN Math. Model. Numer. Anal., 40(1):29-48, 2006.
[Hec12]
F. Hecht. New development in FreeFem++. J. Numer. Math., 20(3-4):251-265, 2012.
[HYZ09]
M. Hinze, N. Yan, and Z. Zhou. Variational discretization for optimal control governed by convection dominated diffusion equations. J. Comput. Math., 27(2-3):237-253, 2009.
[LRL12]
J. Le Rousseau and G. Lebeau. On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations. ESAIM Control Optim. Calc. Var., 18(3):712-747, 2012.
[MS99]
P. Monk and E. Süli. The adaptive computation of far-field patterns by a posteriori error estimation of linear functionals. SIAM J. Numer. Anal., 36(1):251-274, 1999.
[MV12]
E. Malinnikova and S. Vessella. Quantitative uniqueness for elliptic equations with singular lower order terms. Math. Ann., 353(4):1157-1181, 2012.
[YZ09]
N. Yan and Z. Zhou. A priori and a posteriori error analysis of edge stabilization Galerkin method for the optimal control problem governed by convection-dominated diffusion equation. J. Comput. Appl. Math., 223(1):198-217, 2009.
[Zwo12]
M. Zworski. Semiclassical analysis, volume 138 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2012.
[ARRV09] G. Alessandrini, L. Rondi, E. Rosset, and S. Vessella. The stability for the Cauchy problem for elliptic equations. Inverse Problems, 25:123004, 2009. [BHL18] E. Burman, P. Hansbo, and M. G. Larson. Solving ill-posed control problems by stabilized finite element methods: an alternative to Tikhonov regularization. Inverse Problems, 34:035004, 2018. [BNO19a] E. Burman, M. Nechita, and L. Oksanen. A stabilized finite element method for inverse problems subject to the convection-diffusion equation. II: convection-dominated regime. in preparation, 2019. [BNO19b] E. Burman, M. Nechita, and L. Oksanen. Unique continuation for the Helmholtz equation using stabilized finite element methods. J. Math. Pures Appl., 129:1-24, 2019. [BO18] E. Burman and L. Oksanen. Data assimilation for the heat equation using stabilized finite element methods. Numer. Math., 139(3):505-528, 2018. [Bre03] S. C. Brenner. Poincaré-Friedrichs inequalities for piecewise H^(1) functions. SIAM J. Numer. Anal., 41(1):306-324, 2003. [Bur05] E. Burman. A unified analysis for conforming and nonconforming stabilized finite element methods using interior penalty. SIAM J. Numer. Anal., 43(5):2012-2033, 2005. [Bur13] E. Burman. Stabilized finite element methods for nonsymmetric, noncoercive, and ill-posed problems. Part I: elliptic equations. SIAM J. Sci. Comput., 35(6):A2752-A2780, 2013. [Bur14] E. Burman. Error estimates for stabilized finite element methods applied to ill-posed problems. C. R. Math. Acad. Sci. Paris, 352(7-8):655-659, 2014. [BV07] R. Becker and B. Vexler. Optimal control of the convection-diffusion equation using stabilized finite element methods. Numer. Math., 106(3):349-367, 2007. [DQ05] L. Dede' and A. Quarteroni. Optimal control and numerical adaptivity for advectiondiffusion equations. M2AN Math. Model. Numer. Anal., 39(5):1019-1040, 2005. [DSFKSU09] D. Dos Santos Ferreira, C. E. Kenig, M. Salo, and G. Uhlmann. Limiting Carleman weights and anisotropic inverse problems. Invent. Math., 178(1):119-171, 2009. [EG04] A. Ern and J.-L. Guermond. Theory and practice of finite elements, volume 159 of Applied Mathematical Sciences. Springer-Verlag, New York, 2004. [EG06] A. Ern and J.-L. Guermond. Evaluation of the condition number in linear systems arising in finite element approximations. M2AN Math. Model. Numer. Anal., 40(1):29-48, 2006. [Hec12] F. Hecht. New development in FreeFem++. J. Numer. Math., 20(3-4):251-265, 2012. [HYZ09] M. Hinze, N. Yan, and Z. Zhou. Variational discretization for optimal control governed by convection dominated diffusion equations. J. Comput. Math., 27(2-3):237-253, 2009. [LRL12] J. Le Rousseau and G. Lebeau. On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations. ESAIM Control Optim. Calc. Var., 18(3):712-747, 2012. [MS99] P. Monk and E. Süli. The adaptive computation of far-field patterns by a posteriori error estimation of linear functionals. SIAM J. Numer. Anal., 36(1):251-274, 1999. [MV12] E. Malinnikova and S. Vessella. Quantitative uniqueness for elliptic equations with singular lower order terms. Math. Ann., 353(4):1157-1181, 2012. [YZ09] N. Yan and Z. Zhou. A priori and a posteriori error analysis of edge stabilization Galerkin method for the optimal control problem governed by convection-dominated diffusion equation. J. Comput. Appl. Math., 223(1):198-217, 2009. [Zwo12] M. Zworski. Semiclassical analysis, volume 138 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2012.| [ARRV09] | G. Alessandrini, L. Rondi, E. Rosset, and S. Vessella. The stability for the Cauchy problem for elliptic equations. Inverse Problems, 25:123004, 2009. | | :--- | :--- | | [BHL18] | E. Burman, P. Hansbo, and M. G. Larson. Solving ill-posed control problems by stabilized finite element methods: an alternative to Tikhonov regularization. Inverse Problems, 34:035004, 2018. | | [BNO19a] | E. Burman, M. Nechita, and L. Oksanen. A stabilized finite element method for inverse problems subject to the convection-diffusion equation. II: convection-dominated regime. in preparation, 2019. | | [BNO19b] | E. Burman, M. Nechita, and L. Oksanen. Unique continuation for the Helmholtz equation using stabilized finite element methods. J. Math. Pures Appl., 129:1-24, 2019. | | [BO18] | E. Burman and L. Oksanen. Data assimilation for the heat equation using stabilized finite element methods. Numer. Math., 139(3):505-528, 2018. | | [Bre03] | S. C. Brenner. Poincaré-Friedrichs inequalities for piecewise $H^{1}$ functions. SIAM J. Numer. Anal., 41(1):306-324, 2003. | | [Bur05] | E. Burman. A unified analysis for conforming and nonconforming stabilized finite element methods using interior penalty. SIAM J. Numer. Anal., 43(5):2012-2033, 2005. | | [Bur13] | E. Burman. Stabilized finite element methods for nonsymmetric, noncoercive, and ill-posed problems. Part I: elliptic equations. SIAM J. Sci. Comput., 35(6):A2752-A2780, 2013. | | [Bur14] | E. Burman. Error estimates for stabilized finite element methods applied to ill-posed problems. C. R. Math. Acad. Sci. Paris, 352(7-8):655-659, 2014. | | [BV07] | R. Becker and B. Vexler. Optimal control of the convection-diffusion equation using stabilized finite element methods. Numer. Math., 106(3):349-367, 2007. | | [DQ05] | L. Dede' and A. Quarteroni. Optimal control and numerical adaptivity for advectiondiffusion equations. M2AN Math. Model. Numer. Anal., 39(5):1019-1040, 2005. | | [DSFKSU09] | D. Dos Santos Ferreira, C. E. Kenig, M. Salo, and G. Uhlmann. Limiting Carleman weights and anisotropic inverse problems. Invent. Math., 178(1):119-171, 2009. | | [EG04] | A. Ern and J.-L. Guermond. Theory and practice of finite elements, volume 159 of Applied Mathematical Sciences. Springer-Verlag, New York, 2004. | | [EG06] | A. Ern and J.-L. Guermond. Evaluation of the condition number in linear systems arising in finite element approximations. M2AN Math. Model. Numer. Anal., 40(1):29-48, 2006. | | [Hec12] | F. Hecht. New development in FreeFem++. J. Numer. Math., 20(3-4):251-265, 2012. | | [HYZ09] | M. Hinze, N. Yan, and Z. Zhou. Variational discretization for optimal control governed by convection dominated diffusion equations. J. Comput. Math., 27(2-3):237-253, 2009. | | [LRL12] | | | | J. Le Rousseau and G. Lebeau. On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations. ESAIM Control Optim. Calc. Var., 18(3):712-747, 2012. | | [MS99] | P. Monk and E. Süli. The adaptive computation of far-field patterns by a posteriori error estimation of linear functionals. SIAM J. Numer. Anal., 36(1):251-274, 1999. | | [MV12] | E. Malinnikova and S. Vessella. Quantitative uniqueness for elliptic equations with singular lower order terms. Math. Ann., 353(4):1157-1181, 2012. | | [YZ09] | N. Yan and Z. Zhou. A priori and a posteriori error analysis of edge stabilization Galerkin method for the optimal control problem governed by convection-dominated diffusion equation. J. Comput. Appl. Math., 223(1):198-217, 2009. | | [Zwo12] | M. Zworski. Semiclassical analysis, volume 138 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2012. |
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 8, 2019.
