Analysis of divergence-preserving unfitted finite element methods for the mixed Poisson problem

TL;DR

This paper introduces a divergence-preserving unfitted finite element method for the mixed Poisson problem that maintains conservation properties and achieves optimal convergence, even in complex cut configurations.

Contribution

It proposes a new H(div)-conforming unfitted finite element method formulated on the active mesh, ensuring robustness and conservation without pollution from stabilization.

Findings

Method achieves robustness in cut configurations.

Optimal convergence rates for flux and scalar variables.

Numerical experiments confirm theoretical error estimates.

Abstract

In this paper we present a new H(div)-conforming unfitted finite element method for the mixed Poisson problem which is robust in the cut configuration and preserves conservation properties of body-fitted finite element methods. The key is to formulate the divergence-constraint on the active mesh, instead of the physical domain, in order to obtain robustness with respect to cut configurations without the need for a stabilization that pollutes the mass balance. This change in the formulation results in a slight inconsistency, but does not affect the accuracy of the flux variable. By applying post-processings for the scalar variable, in virtue of classical local post-processings in body-fitted methods, we retain optimal convergence rates for both variables and even the superconvergence after post-processing of the scalar variable. We present the method and perform a rigorous a-priori error analysis of the method and discuss several variants and extensions. Numerical experiments confirm the theoretical results.