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arXiv · 2409.20448

On inf-sup stability and optimal convergence of the quasi-reversibility method for unique continuation subject to Poisson's equation

Erik Burman·Mingfei Lu
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Abstract

In this paper, we develop a framework for the discretization of a mixed formulation of quasi-reversibility solutions to ill-posed problems with respect to Poisson's equations. By carefully choosing test and trial spaces a formulation that is stable in a certain residual norm is obtained. Numerical stability and optimal convergence are established based on the conditional stability property of the problem. Tikhonov regularisation is necessary for high order polynomial approximation, , but its weak consistency may be tuned to allow for optimal convergence. For low order elements a simple numerical scheme with optimal convergence is obtained without stabilization. We also provide a guideline for feasible pairs of finite element spaces that satisfy suitable stability and consistency assumptions. Numerical experiments are provided to illustrate the theoretical results.

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Erik Burman, Mingfei Lu. 2024-09-30. On inf-sup stability and optimal convergence of the quasi-reversibility method for unique continuation subject to Poisson's equation. https://arxiv.org/abs/2409.20448

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