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

2024/09/30 by Burman, Erik, Lu, Mingfei
#65N12 #65N20 #65N30 #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2409.20448

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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