A space-time approach to two-phase stokes flow: well-posedness and discretization
Voulis, Igor; Reusken, Arnold (Thesis advisor); Stevenson, Rob (Thesis advisor); Melcher, Christof Erich (Thesis advisor)
Dissertation / PhD Thesis
In this thesis we consider a time-dependent Navier-Stokes two-phase flow. A standard sharp interface model for the fluid dynamics of two-phase flows is studied both from an analytical and a numerical perspective. The Navier-Stokes interface problem has discontinuous density and viscosity coefficients. In such a setting the pressure solution and gradient of the velocity solution are discontinuous across an evolving interface. A closely related linear problem is the two-phase Stokes problem. Despite the fact that this linear Stokes interface problem is a strong simplification of the two-phase Navier-Stokes flow, it is a good model problem for the development of numerical methods. We are particularly interested in a well-posed variational formulation of this Stokes interface problem in a Eulerian setting. We prefer a Eulerian formulation of the Stokes interface problem because we discretize the problem in Euclidean coordinates. Several well-posed formulations are considered. We prove the well-posedness of a variational space-time formulation in suitable spaces of divergence free functions. A variant with a pressure Lagrange multiplier is also considered. With a discontinuous Galerkin (DG) method in mind, we formulate a well-posed discontinuous-in-time version of the problem. The discontinuous-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 methods are discussed in an abstract setting in this thesis. We consider discontinuous Galerkin time discretization methods for abstract parabolic problems with inhomogeneous linear constraints. This includes the Stokes problem with an inhomogeneous (time-dependent) Dirichlet boundary condition and/or an inhomogeneous divergence constraint. Another problem of this kind is the heat equation with an inhomogeneous boundary condition. Two common ways of treating abstract saddle-point problems exist, namely explicit or implicit (via Lagrange multipliers). Therefore, different variational formulations of the parabolic problem with constraints are introduced. For these formulations, different modifications of a standard discontinuous Galerkin time discretization method are considered. Different ways of treating the linear constraints, e.g. ~by using an appropriate projection, are introduced and analyzed. For these discretizations, optimal error bounds, including superconvergence results, are derived. Discretization error bounds for the Lagrange multiplier are presented. Results of experiments confirm the theoretically predicted optimal convergence rates and show that without a modification the (standard) DG method has suboptimal convergence behavior. We consider two explicit examples: the heat equation and the (two-phase) Stokes problem. Fully discrete schemes are discussed in both cases, where the temporal DG scheme is combined with a spatial continuous Galerkin (CG) scheme. For the heat equation we show an optimal error bound with respect to the energy norm. For the Stokes problem a dynamic spatial mesh is considered because it is a useful tool to limit the computational cost for two-phase flow problems where a fine mesh is only necessary near the moving interface. In the case of the one-phase Stokes problem, we show global error bounds which are locally optimal. This is done for the velocity and for the pressure Lagrange multiplier. A space-time scheme for the two-phase Stokes problem is introduced, including a discrete temporal derivative with a discontinuous time-dependent coefficient. Several numerical experiments are performed in the software package DROPS. Standard finite element spaces have a poor approximation quality for discontinuous unknowns. We show the merit of the use of an extended finite element method. This allows us to treat the discontinuity in pressure and gives us an improved method.