Locally conservative immersed finite element method for elliptic interface problems

TL;DR

This paper introduces a locally conservative immersed finite element method (EIFEM) for elliptic interface problems, improving flux conservation and computational efficiency through a novel enrichment and preconditioning approach.

Contribution

The paper presents a new EIFEM with local flux conservation and an auxiliary space preconditioner, addressing limitations of existing methods for interface problems.

Findings

Achieves locally conservative fluxes in EIFEM.

Removes zero eigen-modes via strong Dirichlet boundary conditions.

Numerical tests confirm theoretical advantages.

Abstract

In this paper, we introduce the locally conservative enriched immersed finite element method (EIFEM) to tackle the elliptic problem with interface. The immersed finite element is useful for handling interface with mesh unfit with the interface. However, all the currently available method under IFEM framework may not be designed to consider the flux conservation. We provide an efficient and effective remedy for this issue by introducing a local piecewise constant enrichment, which provides the locally conservative flux. We have also constructed and analyzed an auxiliary space preconditioner for the resulting system based on the application of algebraic multigrid method. The new observation in this work is that by imposing strong Dirichlet boundary condition for the standard IFEM part of EIFEM, we are able to remove the zero eigen-mode of the EIFEM system while still imposing the Dirichlet boundary condition weakly assigned to the piecewise constant enrichment part of EIFEM. A couple of issues relevant to the piecewise constant enrichment given for the mesh unfit to the interface has been discussed and clarified as well. Numerical tests are provided to confirm the theoretical development.