Unique continuation for an elliptic interface problem using unfitted isoparametric finite elements

TL;DR

This paper develops a stabilized unfitted finite element method with isoparametric interface approximation for elliptic interface problems, providing error estimates and numerical validation of unique continuation capabilities.

Contribution

It introduces a novel regularization approach for unfitted finite elements tailored to elliptic interface problems, including error analysis and numerical experiments.

Findings

Regularization improves stability and accuracy.

Interface geometry approximation affects error bounds.

Coefficient heterogeneity influences solution reconstruction.

Abstract

We study unique continuation over an interface using a stabilized unfitted finite element method tailored to the conditional stability of the problem. The interface is approximated using an isoparametric transformation of the background mesh and the corresponding geometrical error is included in our error analysis. To counter possible destabilizing effects caused by non-conformity of the discretization and cope with the interface conditions, we introduce adapted regularization terms. This allows to derive error estimates based on conditional stability. The necessity and effectiveness of the regularization is illustrated in numerical experiments. We also explore numerically the effect of the heterogeneity in the coefficients on the ability to reconstruct the solution outside the data domain. For Helmholtz equations we find that a jump in the flux impacts the stability of the problem significantly less than the size of the wavenumber.