Unique continuation for the Lam\'e system using stabilized finite element methods

TL;DR

This paper presents a stabilized finite element method for solving the unique continuation problem for the time-harmonic elastic wave equation, providing convergence analysis and numerical validation of the approach.

Contribution

It introduces an arbitrary order stabilized finite element method with proven convergence rates for the elastic wave equation's unique continuation problem.

Findings

Convexity of domain influences reconstruction quality.

Higher polynomial orders improve efficiency but increase sensitivity.

Numerical experiments confirm theoretical convergence rates.

Abstract

We introduce an arbitrary order, stabilized finite element method for solving a unique continuation problem subject to the time-harmonic elastic wave equation with variable coefficients. Based on conditional stability estimates we prove convergence rates for the proposed method which take into account the noise level and the polynomial degree. A series of numerical experiments corroborates our theoretical results and explores additional aspects, e.g. how the quality of the reconstruction depends on the geometry of the involved domains. We find that certain convexity properties are crucial to obtain a good recovery of the wave displacement outside the data domain and that higher polynomial orders can be more efficient but also more sensitive to the ill-conditioned nature of the problem.