Computational unique continuation with finite dimensional Neumann trace

TL;DR

This paper develops finite element methods for unique continuation problems involving elliptic equations, providing stability analysis and error estimates when the normal derivative is known in a finite-dimensional space.

Contribution

It introduces a Lipschitz stability result for the unique continuation problem and derives optimal error estimates for a stabilized finite element method.

Findings

Lipschitz stability in the H1-norm for the problem

Optimal a posteriori and a priori error estimates

Effective finite element approximation for unique continuation

Abstract

We consider finite element approximations of unique continuation problems subject to elliptic equations in the case where the normal derivative of the exact solution is known to reside in some finite dimensional space. To give quantitative error estimates we prove Lipschitz stability of the unique continuation problem in the global H1-norm. This stability is then leveraged to derive optimal a posteriori and a priori error estimates for a primal-dual stabilised finite method.