Spacetime finite element methods for control problems subject to the wave equation

TL;DR

This paper develops and analyzes a stabilized spacetime finite element method for solving control problems governed by the wave equation, providing error estimates and numerical validation.

Contribution

It introduces a novel stabilized finite element approach on unstructured spacetime meshes for wave control problems, with rigorous error analysis.

Findings

Error estimates for the approximate control are established.

Numerical experiments confirm the theoretical results.

The method demonstrates stability and accuracy in control computations.

Abstract

We consider the null controllability problem for the wave equation, and analyse a stabilized finite element method formulated on a global, unstructured spacetime mesh. We prove error estimates for the approximate control given by the computational method. The proofs are based on the regularity properties of the control given by the Hilbert Uniqueness Method, together with the stability properties of the numerical scheme. Numerical experiments illustrate the results.