# A time dependent Stokes interface problem: well-posedness and space-time finite element discretization

## TL;DR

This paper develops a well-posed variational formulation for a time-dependent Stokes interface problem with discontinuous coefficients, and introduces a space-time finite element method with numerical validation for two-phase flow simulations.

## Contribution

The paper provides the first well-posed variational formulations for the two-phase Stokes interface problem, including divergence-free and pressure-based variants, and proposes a space-time finite element discretization.

## Key findings

- Well-posed variational formulations are established.
- A space-time finite element method is introduced.
- Numerical experiments demonstrate the method's effectiveness.

## Abstract

In this paper a time dependent Stokes problem that is motivated by a standard sharp interface model for the fluid dynamics of two-phase flows is studied. This Stokes interface problem has discontinuous density and viscosity coefficients and a pressure solution that is discontinuous across an evolving interface. This strongly simplified two-phase Stokes equation is considered to be a good model problem for the development and analysis of finite element discretization methods for two-phase flow problems. In view of the unfitted finite element methods that are often used for two-phase flow simulations, we are particularly interested in a well-posed variational formulation of this Stokes interface problem in a Euclidean setting. Such well-posed weak formulations, which are not known in the literature, are the main results of this paper. Different variants are considered, namely one with suitable spaces of divergence free functions, a discrete-in-time version of it, and variants in which the divergence free constraint in the solution space is treated by a pressure Lagrange multiplier. The discrete-in-time variational formulation involving the pressure variable for the divergence free constraint is a natural starting point for a space-time finite element discretization. Such a method is introduced and results of numerical experiments with this method are presented.