Elliptic Interface Problem approximated by CutFEM: II. A posteriori error analysis based on equilibrated fluxes

TL;DR

This paper develops a reliable and efficient a posteriori error estimator for CutFEM applied to elliptic interface problems with discontinuous coefficients, using equilibrated fluxes to improve accuracy on unfitted meshes.

Contribution

It introduces a novel a posteriori error analysis based on equilibrated fluxes for CutFEM, with sharp reliability and efficiency bounds that account for coefficient discontinuities.

Findings

The error estimator is robust and reliable.

Numerical results confirm the estimator's effectiveness.

Efficiency constants depend explicitly on diffusion coefficients.

Abstract

This paper investigates an elliptic interface problem with discontinuous diffusion coefficients on unfitted meshes, employing the CutFEM method. The main contribution is the a posteriori error analysis based on equilibrated fluxes belonging to the immersed Raviart-Thomas space. We establish sharp reliability and local efficiency of a new error estimator, which includes both volume and interface terms, carefully tracking the dependence of the efficiency constant on the diffusion coefficients and the mesh/interface configuration. Numerical results highlight the robustness of the proposed approach.