Numerical analysis for partial differential equations (PDEs) traditionally considers problems that are well-posed in the continuum, for example the boundary value problem for Poisson’s equation. Computational methods such as the finite element method (FEM) then discretise the problem and provide numerical solutions. However, when a part of the boundary is inaccessible for measurements or no information is given on the boundary at all, the continuum problem might be ill-posed and solving it, in this case, requires regularisation.
In this book we consider the unique continuation problem with (possibly noisy) data given in an interior subset of the domain. This is an ill-posed problem also known as data assimilation and is related to the elliptic Cauchy problem. It arises often in inverse problems and control theory. We will focus on two PDEs for which the stability of this problem depends on the physical parameters: the Helmholtz and the convection–diffusion equations. We first prove conditional stability estimates that are explicit in the wave number and in the Péclet number, respectively, by using Carleman inequalities. Under a geometric convexity assumption, we obtain that for the Helmholtz equation the stability constants grow at most linearly in the wave number.
Then we present a discretise-then-regularise approach for the unique continuation problem. We cast the problem into PDE-constrained optimisation with discrete weakly consistent regularisation. The regularisation is driven by stabilised FEMs and we focus on the interior penalty stabilisation. For the Helmholtz and diffusion-dominated problems, we apply the continuum stability estimates to the approximation error and prove convergence rates by controlling the residual through stabilisation. For convection dominated problems, we perform a different error analysis and obtain sharper weighted error estimates along the characteristics of the convective field through the data region, with quasi-optimal convergence rates. The results are illustrated by numerical examples.
1 Ill-posed inverse problems and unique continuation
1.1 Well-posed problems
1.2 Linear inverse problems
1.2.1 Conditionally stable problems
1.3 The Cauchy problem
1.4 Unique continuation
2 Conditional stability estimates for unique continuation
2.1 A pointwise Carleman estimate for the Laplacian
2.2 Holder stability estimates
2.3 Convection–diffusion operator
2.3.1 Shifting the norms
2.4 Helmholtz operator
2.4.1 Shifting the norms
3 Discrete regularisation using stabilised finite element methods
3.2 Stabilised finite element methods
3.3 Continuous interior penalty
4 Helmholtz equation
4.2 Error estimates
4.2.1 Data perturbations
4.3 Numerical examples
5 Diffusion-dominated problems
5.2 Error estimates
5.3 Numerical examples
6 Convection-dominated problems
6.1.1 Stability region and weight functions
6.1.2 Discrete commutator property
6.2 Error estimates
6.2.1 Downstream estimates
6.2.2 Upstream estimates
6.3 Numerical examples
A Finite element inequalities
B Pseudodifferential operators
unique continuation, ill-posed problems, stabilised finite element methods, data assimilation
M. Nechita, Unique continuation problems and stabilised finite element methods, Casa Cărții de Știință, Cluj-Napoca, Romania, 2021, ISBN: 978-606-17-1816-0
Unique continuation problems and stabilised finite element methods
Casa Cărții de Știință
The book on google scholar.
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