Optimal preconditioners for Nitsche-XFEM discretizations of interface problems

TL;DR

This paper develops and analyzes optimal preconditioners for Nitsche-XFEM discretizations of interface problems, ensuring condition numbers are independent of mesh size and interface position, with supporting numerical experiments.

Contribution

It introduces an additive subspace preconditioner that is optimal and independent of mesh size and interface location for Nitsche-XFEM methods.

Findings

Preconditioner achieves mesh-independent condition numbers.

Diagonal scaling bounds the condition number by ch^{-2}.

Numerical experiments confirm theoretical results.

Abstract

In the past decade, a combination of unfitted finite elements (or XFEM) with the Nitsche method has become a popular discretization method for elliptic interface problems. This development started with the introduction and analysis of this Nitsche-XFEM technique in the paper [A. Hansbo, P. Hansbo, Comput. Methods Appl. Mech. Engrg. 191 (2002)]. In general, the resulting linear systems have very large condition numbers, which depend not only on the mesh size $h$, but also on how the interface intersects the mesh. This paper is concerned with the design and analysis of optimal preconditioners for such linear systems. We propose an additive subspace preconditioner which is optimal in the sense that the resulting condition number is independent of the mesh size $h$ and the interface position. We further show that already the simple diagonal scaling of the stifness matrix results in a condition number that is bounded by $ch^{-2}$, with a constant $c$ that does not depend on the location of the interface. Both results are proven for the two-dimensional case. Results of numerical experiments in two and three dimensions are presented, which illustrate the quality of the preconditioner.