Unique continuation for the wave equation: the stability landscape

TL;DR

This paper investigates unique continuation for the wave equation, establishing stability results that enable the design of convergent finite element methods based on the problem's stability landscape.

Contribution

It provides new stability estimates for wave equation continuation from partial data, facilitating the development of convergent numerical schemes.

Findings

Hölder stability for continuation into a subset of the domain

Lipschitz stability with finite boundary data

Finite element method convergence matching stability properties

Abstract

We consider a unique continuation problem for the wave equation given data in a volumetric subset of the space time domain. In the absence of data on the lateral boundary of the space-time cylinder we prove that the solution can be continued with H\"older stability into a certain proper subset of the space-time domain. Additionally, we show that unique continuation of the solution to the entire space-time cylinder with Lipschitz stability is possible given the knowledge of a suitable finite dimensional space in which the trace of the solution on the lateral boundary is contained. These results allow us to design a finite element method that provably converges to the exact solution at a rate that mirrors the stability properties of the continuous problem.