High Order Cut Finite Element Methods for the Stokes Problem

TL;DR

This paper introduces a high order cut finite element method for the Stokes problem that ensures stability and optimal error estimates on composite meshes with overlapping regions.

Contribution

It develops a novel high order cut finite element approach for the Stokes problem using Nitsche formulation and stabilization, ensuring inf-sup stability on composite meshes.

Findings

The method achieves inf-sup stability on composite meshes.

Optimal a priori error estimates are proven.

The approach is applicable to general inf-sup stable spaces.

Abstract

We develop a high order cut finite element method for the Stokes problem based on general inf-sup stable finite element spaces. We focus in particular on composite meshes consisting of one mesh that overlaps another. The method is based on a Nitsche formulation of the interface condition together with a stabilization term. Starting from inf-sup stable spaces on the two meshes, we prove that the resulting composite method is indeed inf-sup stable and as a consequence optimal \emph{a~priori} error estimates hold.