A stabilized Nitsche fictitious domain method for the Stokes problem

TL;DR

This paper introduces a stabilized Nitsche fictitious domain method for the Stokes problem that ensures stability, optimal error estimates, and boundary-agnostic condition numbers, supported by theoretical analysis and numerical experiments.

Contribution

It develops a new stabilized Nitsche fictitious domain method for the Stokes problem with proven stability and optimal error estimates, applicable in three dimensions.

Findings

The method is inf-sup stable.

Optimal order a priori error estimates are established.

The condition number is independent of boundary location.

Abstract

We develop a Nitsche fictitious domain method for the Stokes problem starting from a stabilized Galerkin finite element method with low order elements for both the velocity and the pressure. By introducing additional penalty terms for the jumps in the normal velocity and pressure gradients in the vicinity of the boundary, we show that the method is inf-sup stable. As a consequence, optimal order a priori error estimates are established. Moreover, the condition number of the resulting stiffness matrix is shown to be bounded independently of the location of the boundary. We discuss a general, flexible and freely available implementation of the method in three spatial dimensions and present numerical examples supporting the theoretical results.