# New stability estimates for an unfitted finite element method for two-phase Stokes problem

## TL;DR

This paper develops new stability estimates for an unfitted finite element method applied to the two-phase Stokes problem, ensuring robustness with respect to viscosity jumps and providing theoretical and numerical validation.

## Contribution

It introduces stability estimates for an unfitted finite element approach to the two-phase Stokes problem, with error bounds independent of viscosity variations.

## Key findings

- Stability constants are independent of viscosity coefficients.
- Finite element error estimates are robust against interface misalignment.
- Numerical experiments confirm theoretical stability and accuracy.

## Abstract

The paper addresses stability and finite element analysis of the stationary two-phase Stokes problem with a piecewise constant viscosity coefficient experiencing a jump across the interface between two fluid phases. We first prove a priori estimates for the individual terms of the Cauchy stress tensor with stability constants independent of the viscosity coefficient. Next, this stability result is extended to the approximation of the two-phase Stokes problem by a finite element method. In the method considered, the interface between the phases does not respect the underlying triangulation, which put the finite element method into the class of unfitted discretizations. The finite element error estimates are proved with constants independent of viscosity. Numerical experiments supporting the theoretical results are provided.