Parallel multilevel solvers for coupled interface problems

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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 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 decomposition approaches or multilevel methods. The parallel treatment of constrained or heterogeneous problems, 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., droplets impacting on a wall (liquid-gas flow) 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 project 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 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 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 and crack propagation, both applications lead to large saddle point problems, additionally complicated by the numerical treatment of time dependent interfaces. 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.