Multilevel preconditioning of stabilized unfitted finite element discretizations

Ludescher, Thomas; Reusken, Arnold (Thesis advisor); Lehrenfeld, Christoph (Thesis advisor)

Aachen (2020)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2020

Abstract

In this thesis new methods for the iterative solution of interface problems are presented. Two problem classes are considered: the Poisson interface problem and the Stokes interface problem. These are model problems for mass transport and hydrodynamics in a two-phase flow system. For realistic simulations of these systems one typically has to deal with an evolving interface. In this work we only consider quasi-stationary problems with a \emph{stationary} interface. To mimic the setting of a realistic problem we use the level set method for the interface description. In that setting the mesh is \emph{not} aligned to the interface. Additionally, the model problems exhibit discontinuities in the solution across the interface, when the material coefficients are discontinuous or (for the Stokes case) a surface tension force is applied. When applying an iterative solver to these model problems, critical parameters are the mesh size, location of the interface and large contrast in material coefficients. \\ \noindentA stable and accurate discretization is an essential basis for designing a robust iterative solver. To approximate discontinuities within mesh elements we use an unfitted finite element method. The interface conditions, however, are not part of the solution space. Therefore, we use a Nitsche technique to weakly impose those conditions. Within Nitsche's method one has some freedom in the choice of averaging weights. We choose those weights such that we obtain robustness with respect to large contrast in material parameters. To obtain robustness with respect to the location of the interface, we use a ghost penalty stabilization technique. In case of the Stokes problem, the stability of the method depends on the pair of spaces used for velocity and pressure. We use a $\cP_2$--$\cP_1$ Taylor-Hood pair, which is stable for fitted finite element methods, and apply an additional ghost penalty term for the pressure to obtain robustness in the unfitted case. Typically, a higher order accuracy of the discretization is limited by a piecewise linear approximation of the interface. To gain a higher order geometry approximation we use an isoparametric mapping. We analyze the stability properties of the discretizations for both model problems and demonstrate these by numerical experiments. Furthermore, we provide discretization error bounds for the Poisson interface problem and study their dependence on material coefficients. Numerical experiments confirm the theoretical findings and show optimal order convergence for both model problems. \\\noindentFor the Poisson interface problem, we develop a new multigrid method. Transfer operators for the unfitted finite element spaces on different levels are designed, which are robust with respect to the location of the interface. Furthermore, an interface smoother is developed to yield robustness with respect to discontinuous material coefficients. We present a convergence analysis for this method, which provides a bound that is independent of the mesh size and the location of the interface. Numerical experiments show robustness of the method with respect to mesh size, location of the interface and contrast in material coefficients. The method is extended to higher order discretizations using a two-grid approach. \\\noindentFor the Stokes interface problem we use a Cahout-Chabard-type Schur complement preconditioner, which consists of the solution of a scaled mass and Poisson problem on the pressure space. For preconditioning of the Poisson part we can use the coefficient robust discretization introduced previously, for which the new multigrid method can be applied. Also appropriate boundary conditions for the preconditioner are discussed. The velocity block discretization is designed such that it has the same structure as a vector valued Poisson interface problem and therefore the higher order multigrid method can be applied. Numerical experiments illustrate the robustness of the proposed preconditioner with respect to mesh size, location of the interface and contrast in material coefficients. \\\noindentThe aforementioned methods have been implemented partially in the \texttt{DROPS} software package, as well as in the \texttt{ngsxfem} package as an add-on to the \texttt{ngsolve} software package.