Arnold Reusken

An extended pressure finite element space for two-phase incompressible flows with surface tension

We consider a standard model for incompressible two-phase flows in which a localized force at the interface describes the effect of surface tension. If a level set (or VOF) method is applied then the interface, which is implicitly given by the zero level of the level set function, is in general not aligned with the triangulation that is used in the discretization of the flow problem. This non-alignment causes severe difficulties w.r.t. the discretization of the localized surface tension force and the discretization of the flow variables. In cases with large surface tension forces the pressure has a large jump across the interface. In standard finite element spaces, due to the non-alignment, the functions are continuous across the interface and thus not appropriate for the approximation of the discontinuous pressure. In many simulations these effects cause large oscillations of the velocity close to the interface, so-called spurious velocities. In this paper, for a simplified model problem, we give an analysis that explains why known (standard) methods for discretization of the localized force term and for discretization of the pressure variable often yield large spurious velocities. In the paper [S. Grosz, A. Reusken, Finite element discretization error analysis of a surface tension force in two-phase incompressible flows, Preprint 262, IGPM, RWTH Aachen, SIAM J. Numer. Anal. (accepted for publication)], we introduce a new and accurate method for approximation of the surface tension force. In the present paper, we use the extended finite element space (XFEM), presented in [N. Moes, J. Dolbow, T. Belytschko, A finite element method for crack growth without remeshing, Int. J. Numer. Meth. Eng. 46 (1999) 131-150; T. Belytschko, N. Moes, S. Usui, C. Parimi, Arbitrary discontinuities in finite elements, Int. J. Numer. Meth. Eng. 50 (2001) 993-1013], for the discretization of the pressure. We show that the size of spurious velocities is reduced substantially, provided we use both the new treatment of the surface tension force and the extended pressure finite element space.

Numerical methods for two-phase incompressible flows

Introduction.- Part I One-phase incompressible flows.- Mathematical models.- Finite element discretization.- Time integration.-

Grad-div stablilization for Stokes equations

In this paper a stabilizing augmented Lagrangian technique for the Stokes equations is studied. The method is consistent and hence does not change the continuous solution. We show that this stabilization improves the well-posedness of the continuous problem for small values of the viscosity coefficient. We analyze the influence of this stabilization on the accuracy of the finite element solution and on the convergence properties of the inexact Uzawa method.

A Finite Element Method for Elliptic Equations on Surfaces

In this paper a new finite element approach for the discretization of elliptic partial differential equations on surfaces is treated. The main idea is to use finite element spaces that are induced by triangulations of an “outer” domain to discretize the partial differential equation on the surface. The method is particularly suitable for problems in which there is a coupling with a flow problem in an outer domain that contains the surface. We give an analysis that shows that the method has optimal order of convergence both in the

Analysis of a damped nonlinear multilevel method

H^1

Finite Element Discretization Error Analysis of a Surface Tension Force in Two-Phase Incompressible Flows

- and in the

Uniform preconditioners for a parameter dependent saddle point problem with application to generalized Stokes interface equations

L^2

Analysis of a Stokes interface problem

-norm. Results of numerical experiments are included that confirm this optimality.

Robust Parallel Smoothing for Multigrid Via Sparse Approximate Inverses

SummaryIn this paper, we present a new algorithm that is obtained by introducing a damping parameter in the classical Nonlinear Multilevel Method. We analyse this Damped Nonlinear Multilevel Method. In particular, we prove global convergence and local efficiency for a suitable class of problems.

Fast Iterative Solvers for Discrete Stokes Equations

We consider a standard model for a stationary two-phase incompressible flow with surface tension. In the variational formulation of the model a linear functional which describes the surface tension force occurs. This functional depends on the location and the curvature of the interface. In a finite element discretization method the functional has to be approximated. For an approximation method based on a Laplace-Beltrami representation of the curvature we derive sharp bounds for the approximation error. A new modified approximation method with a significantly smaller error is introduced.