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IGPM479.pdf March 2018 
TITLE 
A Time Dependent Stokes Interface Problem: WellPosedness and SpaceTime Finite Element Discretization 
AUTHORS 
Igor Voulis, Arnold Reusken 
ABSTRACT 
In this paper a time dependent Stokes problem that is motivated by a standard sharp interface model for the fluid dynamics of twophase 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 twophase Stokes equation is considered to be a good model problem for the development and analysis of finite element discretization methods for twophase flow problems. In view of the
unfitted finite element methods that are often used for twophase flow simulations, we are particularly interested in a wellposed variational formulation of this Stokes interface problem in a Euclidean setting. Such wellposed 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 discreteintime version of it, and variants in which the divergence free constraint in the solution space is treated by a pressure Lagrange multiplier. The discreteintime variational formulation involving the pressure variable for the divergence free constraint is a natural starting point for a spacetime finit e element discretization. Such a method is introduced and results of a numerical experiment with this method are presented. 
KEYWORDS 
spacetime variational saddle point formulation, wellposed operator equation, twophase flow, XFEM, spacetime finite element method, discontinuous galerkin, fourdimensional computations 