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

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

Discontinuous Galerkin Time Discretization Methods for Parabolic Problems with Linear Constraints

We consider time discretization methods for abstract parabolic problems with inhomogeneous linear constraints. Prototype examples that fit into the general framework are the heat equation with inhomogeneous (time dependent) Dirichlet boundary conditions and the time dependent Stokes equation with an inhomogeneous divergence constraint. Two common ways of treating such linear constraints, namely explicit or implicit (via Lagrange multipliers) are studied. These different treatments lead to different variational formulations of the parabolic problem. For these formulations we introduce a modification of the standard discontinuous Galerkin (DG) time discretization method in which an appropriate projection is used in the discretization of the constraint. 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 the modification the (standard) DG method has sub-optimal convergence behavior.Comment: 35 page

Analysis of divergence-preserving unfitted finite element methods for the mixed Poisson problem

In this paper we present a new H(div)-conforming unfitted finite element method for the mixed Poisson problem which is robust in the cut configuration and preserves conservation properties of body-fitted finite element methods. The key is to formulate the divergence-constraint on the active mesh, instead of the physical domain, in order to obtain robustness with respect to cut configurations without the need for a stabilization that pollutes the mass balance. This change in the formulation results in a slight inconsistency, but does not affect the accuracy of the flux variable. By applying post-processings for the scalar variable, in virtue of classical local post-processings in body-fitted methods, we retain optimal convergence rates for both variables and even the superconvergence after post-processing of the scalar variable. We present the method and perform a rigorous a-priori error analysis of the method and discuss several variants and extensions. Numerical experiments confirm the theoretical results.Comment: 26 pages, 6 figure

A space-time approach to two-phase stokes flow: well-posedness and discretization

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

An adaptive stochastic Galerkin method based on multilevel expansions of random fields: Convergence and optimality

The subject of this work is a new stochastic Galerkin method for second-order elliptic partial differential equations with random diffusion coefficients. It combines operator compression in the stochastic variables with tree-based spline wavelet approximation in the spatial variables. Relying on a multilevel expansion of the given random diffusion coefficient, the method is shown to achieve optimal computational complexity up to a logarithmic factor. In contrast to existing results, this holds in particular when the achievable convergence rate is limited by the regularity of the random field, rather than by the spatial approximation order. The convergence and complexity estimates are illustrated by numerical experiments