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

In this paper, we develop a framework for the discretization of a mixed formulation of quasi-reversibility methods for 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 regularization 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 assumptions. Numerical experiments are provided to illustrate the theoretical results.