On inf-sup stability and optimal convergence of the quasi-reversibility method for unique continuation subject to Poisson's equation
<|reference_start|>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 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.<|reference_end|>
@article{burman2024on,
    title={On inf-sup stability and optimal convergence of the quasi-reversibility
  method for unique continuation subject to Poisson's equation},
    author={Erik Burman and Mingfei Lu},
    journal={arXiv preprint arXiv:2409.20448},
    year={2024},
    archivePrefix={arXiv},
    eprint={2409.20448},
    primaryClass={math.NA cs.NA}
}