Conservative flux reconstruction for an elliptic interface problem using CutFEM

TL;DR

This paper introduces a novel flux reconstruction method for elliptic interface problems with discontinuous coefficients, ensuring flux continuity and conservation using CutFEM and immersed Raviart-Thomas spaces.

Contribution

It proposes a new flux recovery approach that guarantees flux continuity and conservation across interfaces in unfitted finite element discretizations.

Findings

The immersed Raviart-Thomas flux ensures normal flux continuity across the interface.

The recovered flux leads to a reliable a posteriori error estimator.

The method achieves sharp error bounds in the numerical analysis.

Abstract

This paper deals with the local recovery of conservative fluxes for an elliptic interface problem with discontinuous coefficients. The transmission conditions on the interface are imposed weakly and the discretisation is achieved by using conforming finite elements on unfitted meshes, with the aid of the CutFEM method. In a first attempt at flux reconstruction, we define a flux belonging to the Raviart-Thomas space of each sub-domain following the method developed for a boundary problem. However, the transmission condition is not satisfied by the recovered flux. In order to overcome this shortcoming, we propose a second approach, where the flux belongs to the recently introduced immersed Raviart-Thomas space. This ensures both the continuity of the normal flux across the interface and a natural conservation property on the cut cells. Subsequently, we apply the recovered flux to a posteriori error analysis and prove the sharp reliability of the error estimator.