Parallel multilevel solvers for coupled interface problems

Although during the last decades tremendous progress has been achieved in the area of parallel finite element simulations, the parallel solution of complex and constrained problems in, e. g., mechanics and fluid mechanics, still remains a challenging task. Usually, good scalability can be achieved relatively straightforwardly for homogeneous problems, i. e., problems with smooth data, and structured meshes. Parallel simulations involving unstructured and adaptive meshes have also been successfully carried out, using a wide variety of solution methods, ranging from Krylov subspace methods to domain decomposi- tion approaches or multilevel methods. The parallel treatment of constrained or strongly heterogeneous problems, i. e., problems with very rough data, however, is still far from trivial. This is caused by the more complex mathematical structure of the discrete systems to be solved and by the more advanced data structures employed for assembling, solving, and parallel data exchange.

An additional and up to now only scarcely addressed difficulty arises if time-dependent interfaces have to be resolved, as is the case in, e.g., liquid-gas flows of droplets impacting on a wall or crack propagation. These interfaces do not only influence the discretization, but also the robustness of iterative solution methods. Moreover, the special treatment of the interface poses a challenge for the parallel distribution of geometric objects across large scale machines.

Efficient solution methods for these problems and their scalable and flexible implementations put a high demand not only on underlying methodology but also on the software used. In this proposal, we therefore aim at the development and implementation of both, fast solution methods as well as efficient software tools. More precisely, we will develop and implement parallel multilevel solvers for saddle point problems which are able to deal with time-dependent interfaces in a robust manner. To this end, we will employ a new approach for the construction of multilevel hierarchies, which is based on non-standard transfer (i.e., restriction and interpolation) operators and solution-dependent coarse grid spaces. As a basis for this work, one central goal of this project is the development and implementation of a stand-alone high-level library for the parallel management of distributed geometric objects. In order to foster the broad applicability of this library, we will use it within two different software and application environments, namely DROPS at RWTH and ObsLib++ at USI. Despite the seemingly different application fields, RWTH with two-phase flow and USI with computational mechanics, contact and crack propagation, both applications lead to large saddle point problems, with the additional complication of time-dependent interfaces. As these interfaces are typically only resolved by the finest mesh of the multilevel hierarchy and not by its coarse meshes, the multilevel solvers and their parallelization have to be adapted carefully. This will be done by adapting the intergrid operators as well as the ansatz spaces on each level. This combined approach is expected not only to provide the necessary robustness of the solver but also to allow for clear and modular data structures, simplifying the parallel implementation without sacrificing scalability.

For this project strong expertise in the handling of time-dependent interfacial constraints, multilevel methods and solvers for saddle point problems, non-standard transfer operators, and HPC software for efficient parallel data management is mandatory. The applicants at RWTH and USI together provide in a complementary manner this expertise, which motivated the formation of this research team and the Swiss-German cooperation.