Space time stabilized finite element methods for a unique continuation problem subject to the wave equation

TL;DR

This paper introduces a stabilized spacetime finite element method for solving a unique continuation problem for the wave equation, achieving optimal convergence and validated through numerical examples.

Contribution

It develops a primal-dual stabilized finite element approach on unstructured spacetime meshes for wave equations, with proven error estimates and optimal convergence.

Findings

Error estimates based on stability and geometric control condition

Optimal order of convergence achieved

Numerical examples confirm theoretical results

Abstract

We consider a stabilized finite element method based on a spacetime formulation, where the equations are solved on a global (unstructured) spacetime mesh. A unique continuation problem for the wave equation is considered, where data is known in an interior subset of spacetime. For this problem, we consider a primal-dual discrete formulation of the continuum problem with the addition of stabilization terms that are designed with the goal of minimizing the numerical errors. We prove error estimates using the stability properties of the numerical scheme and a continuum observability estimate, based on the sharp geometric control condition by Bardos, Lebeau and Rauch. The order of convergence for our numerical scheme is optimal with respect to stability properties of the continuum problem and the interpolation errors of approximating with polynomial spaces. Numerical examples are provided that illustrate the methodology.