Parallel multilevel solvers for coupled interface problems
Final Report Abstract
For the development of a parallel finite element software for an efficient simulation of coupled interface problems in structural and fluid mechanics various requirements have to be fulfilled. Firstly, accurate and robust discretization methods are needed. Due to the (evolving) interfaces this typically leads to non-standard finite element methods (such as XFEM or CutFEM). Secondly, multilevel grid structures have to be used to accurately resolve the interface and to apply optimal multilevel solvers. Finally the parallelization of such multilevel solution algorithms requires the management of complex parallel data structures to store the distributed multilevel grid, a dynamic load balancing strategy to account for the time-dependent refinement zone and the ability for efficient communication across processor boundaries. Concerning this parallelization aspect, all these requirements are met by the DiST library. In this project this library was generalized to more general non-tetrahedral meshes and made available to the research partner at USI. Regarding the issue of accurate and robust discretizations, the ghost penalty stabilization for extended finite element discretizations (XFEM) has been thoroughly studied for a class of two-phase Stokes problems. Stability aspects of this method are theoretically investigated, the method was implemented in the DROPS solver and tested for several applications, including two-phase flows described by the Navier-Stokes equations. Concerning robust and efficient (multilevel) iterative solvers, several topics were studied in the project. A robust Schur complement preconditioner for the XFEM discretization with ghost-penalty stabilization has been developed, analyzed and implemented. The structure of this preconditioner allows an easy parallelization in the framework of parallel preconditioned Krylov solvers. A further topic was the development of efficient multigrid solvers for scalar elliptic interface problems and for Stokes interface problems. It turned out that both new transfer operators and improved smoothers were needed to obtain a multigrid solver that was efficient also for cases with large differences in density and viscosity coefficients in the two phases. A new general strategy for constructing suitable transfer operators for unfitted finite elements was introduced. A tailor-made robust smoother for this specific problem class has been developed. These enhanced discretization methods for the incompressible twophase Stokes problem combined with the newly developed multigrid method and the DiST library for parallelization are key ingredients for the efficient simulation of coupled interface problems in structural and fluid mechanics. We briefly address a few topics that are currently investigated. So far the XFEM technique was applied only for the discretization of the pressure variable (which is discontinuous across the interface). To obtain an accurate discretization also in the case of large viscosity ratios (as in liquid-gas two-phase systems) the velocity discretization has to be improved. For this we plan to use a Nitsche-XFEM discretization to enable the accurate representation of kinks at the interface. This improved discretization will then also require (minor) modifications of the newly developed multigrid solver. A further research topic is the development of a multigrid solver for the fully coupled Stokes interface problem. For this the already available transfer operators can be used, but a new Vanka type smoother has to be designed.
Publications
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Analysis of an XFEM discretization for Stokes interface problems. SIAM Journal on Scientific Computing, 38(2):A1019–A1043, 2016
M. Kirchhart, S. Gross, and A. Reusken
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Robust preconditioning for XFEM applied to timedependent Stokes problems. SIAM Journal on Scientific Computing, 38(6):A3492–A3514, 2016
S. Gross, T. Ludescher, M. Olshanskii, and A. Reusken
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A multigrid method for unfitted finite element discretizations of elliptic interface problems
T. Ludescher, S. Gross and A. Reusken
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Efficient multilevel solvers for stabilized CutFEM discretizations. PhD thesis, completed in 2019
T. Ludescher
