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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.
Voulis, Igor 1 ; Reusken, Arnold 1
@article{M2AN_2018__52_6_2187_0,
author = {Voulis, Igor and Reusken, Arnold},
title = {A time dependent {Stokes} interface problem: well-posedness and space-time finite element discretization},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {2187--2213},
publisher = {EDP-Sciences},
volume = {52},
number = {6},
year = {2018},
doi = {10.1051/m2an/2018053},
mrnumber = {3905193},
zbl = {1414.76037},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2018053/}
}
TY - JOUR AU - Voulis, Igor AU - Reusken, Arnold TI - A time dependent Stokes interface problem: well-posedness and space-time finite element discretization JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 2187 EP - 2213 VL - 52 IS - 6 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an/2018053/ DO - 10.1051/m2an/2018053 LA - en ID - M2AN_2018__52_6_2187_0 ER -
%0 Journal Article %A Voulis, Igor %A Reusken, Arnold %T A time dependent Stokes interface problem: well-posedness and space-time finite element discretization %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 2187-2213 %V 52 %N 6 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/m2an/2018053/ %R 10.1051/m2an/2018053 %G en %F M2AN_2018__52_6_2187_0
Voulis, Igor; Reusken, Arnold. A time dependent Stokes interface problem: well-posedness and space-time finite element discretization. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 6, pp. 2187-2213. doi: 10.1051/m2an/2018053
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