A Multigrid Method for Unfitted Finite Element Discretizations of Elliptic Interface Problems

TL;DR

This paper develops a multigrid solver for unfitted finite element methods applied to elliptic interface problems, demonstrating efficiency and robustness across various challenging scenarios.

Contribution

It introduces a novel multigrid approach with a new prolongation operator and interface smoother tailored for unfitted finite element discretizations.

Findings

Efficient multigrid convergence demonstrated in numerical tests.

Robustness against large coefficient jumps and mesh variations.

Effective handling of interface location and mesh size changes.

Abstract

We consider discrete Poisson interface problems resulting from linear unfitted finite elements, also called cut finite elements (CutFEM). Three of these unfitted finite element methods known from the literature are studied. All three methods rely on Nitsche s method to incorporate the interface conditions. The main topic of the paper is the development of a multigrid method, based on a novel prolongation operator for the unfitted finite element space and an interface smoother that is designed to yield robustness for large jumps in the diffusion coefficients. Numerical results are presented which illustrate efficiency of this multigrid method and demonstrate its robustness properties with respect to variation of the mesh size, location of the interface and contrast in the diffusion coefficients.