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

TL;DR

This paper develops a stable and optimally convergent mixed finite element framework for quasi-reversibility solutions to ill-posed Poisson problems, with practical guidelines and numerical validation.

Contribution

It introduces a new discretization framework ensuring stability and optimal convergence for quasi-reversibility methods applied to Poisson's equation.

Findings

Stable discretization with optimal convergence rates.

Tikhonov regularization improves high-order polynomial approximation.

Numerical experiments confirm theoretical stability and convergence.

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.