Unique continuation for the wave equation based on a discontinuous Galerkin time discretization

TL;DR

This paper introduces a stable method for reconstructing wave equation solutions with missing initial data using a discontinuous Galerkin time discretization combined with finite elements, supported by error estimates and preconditioning strategies.

Contribution

It presents a novel approach employing standard discontinuous Galerkin methods for stable wave solution reconstruction under geometric control conditions.

Findings

Error estimates are established under geometric control.

Preconditioning strategies improve solution efficiency.

Numerical experiments confirm the importance of geometric control.

Abstract

We consider a stable unique continuation problem for the wave equation where the initial data is lacking and the solution is reconstructed using measurements in some subset of the bulk domain. Typically fairly sophisticated space-time methods have been used in previous work to obtain stable and accurate solutions to this reconstruction problem. Here we propose to solve the problem using a standard discontinuous Galerkin method for the temporal discretization and continuous finite elements for the space discretization. Error estimates are established under a geometric control condition. We also investigate two preconditioning strategies which can be used to solve the arising globally coupled space-time system by means of simple time-stepping procedures. Our numerical experiments test the performance of these strategies and highlight the importance of the geometric control condition for reconstructing the solution beyond the data domain.