An unfitted finite element method for two-phase Stokes problems with slip between phases

TL;DR

This paper introduces a stable unfitted finite element method for simulating two-phase Stokes flow with slip, demonstrating robustness and optimal error estimates independent of interface position and material contrast.

Contribution

It develops a novel unfitted finite element approach with proven stability and error bounds for two-phase Stokes problems involving slip between phases.

Findings

Method is stable regardless of interface position and viscosity contrast.

Numerical results confirm theoretical stability and accuracy.

Approach is effective in both 2D and 3D simulations.

Abstract

We present an isoparametric unfitted finite element approach of the CutFEM or Nitsche-XFEM family for the simulation of two-phase Stokes problems with slip between phases. For the unfitted generalized Taylor--Hood finite element pair $\mathbf{P}_{k+1}-P_k$, $k\ge1$, we show an inf-sup stability property with a stability constant that is independent of the viscosity ratio, slip coefficient, position of the interface with respect to the background mesh and, of course, mesh size. In addition, we prove stability and optimal error estimates that follow from this inf-sup property. We provide numerical results in two and three dimensions to corroborate the theoretical findings and demonstrate the robustness of our approach with respect to the contrast in viscosity, slip coefficient value, and position of the interface relative to the fixed computational mesh.