Higher order unfitted FEM for Stokes interface problems

TL;DR

This paper introduces a higher order unfitted finite element method for solving Stokes interface problems with implicit geometries, combining Nitsche's method, ghost penalty stabilization, and advanced geometry approximation techniques.

Contribution

It develops a novel higher order unfitted FEM approach for Stokes interface problems that handles implicit geometries and discontinuities accurately and efficiently.

Findings

The method achieves stable and accurate solutions for complex interface problems.

Numerical examples demonstrate the effectiveness of the approach.

The approach handles strong and weak discontinuities across interfaces.

Abstract

We consider the discretization of a stationary Stokes interface problem in a velocity-pressure formulation. The interface is described implicitly as the zero level of a scalar function as it is common in level set based methods. Hence, the interface is not aligned with the mesh. An unfitted finite element discretization based on a Taylor-Hood velocity-pressure pair and an XFEM (or CutFEM) modification is used for the approximation of the solution. This allows for the accurate approximation of solutions which have strong or weak discontinuities across interfaces which are not aligned with the mesh. To arrive at a consistent, stable and accurate formulation we require several additional techniques. First, a Nitsche-type formulation is used to implement interface conditions in a weak sense. Secondly, we use the ghost penalty stabilization to obtain an inf-sup stable variational formulation. Finally, for the highly accurate approximation of the implicitly described geometry, we use a combination of a piecewise linear interface reconstruction and a parametric mapping of the underlying mesh. We introduce the method and discuss results of numerical examples.