Adaptive FETI-DP and BDDC methods for highly heterogeneous elliptic finite element problems in three dimensions

Numerical methods are often well-suited for the solution of (elliptic) partial differential equations (PDEs) modeling naturally occuring processes. Many different solvers can be applied to systems which are obtained after discretization by the finite element method. Parallel architectures in modern computers facilitate the efficient use of diverse divide and conquer strategies. The intuitive approach, to divide a large (global) problem into subproblems, which are then solved in parallel, can significantly reduce the solution time. It is obvious that the solvers on the local subproblems then should deliver the contributions of the global solution restricted to the subdomains of computational region. The class of domain decomposition methods provides widely-used iterative algorithms for the parallel solution of implicit finite element problems. Often, an additional coarse space, which introduces a coupling between the subdomains, is used to ensure a global transport of information between the subdomains across the entire domain. The FETI-DP and BDDC domain decomposition methods are highly scalable parallel algorithms. However, when the parameter or coefficient distribution in the underlying partial differential equation becomes highly heterogeneous, classical methods, with a priori chosen coarse spaces, might not converge in a limited number of iterations. A remedy is offered by problem-dependent coarse spaces. These coarse spaces can be provided by adaptive methods, which then can improve the convergence at the cost of additional constraints. In this thesis, we introduce robust FETI-DP and BDDC methods for three-dimensional problems. These methods incorporate constraints, which are computed from local eigenvalue problems on faces and edges between subdomains, into the coarse space. The implementation of the constraints is performed by a deflation or balancing approach or by partial finite element assembly after a transformation of basis. For the latter, we introduce the generalized transformation-of-basis approach and show its correspondence to a deflation or balancing approach. An efficient parallel implementation of adaptive FETI-DP is discussed in the last part of this thesis. We provide weak and strong parallel scalability results for our adaptive algorithm executed on the supercomputer magnitUDE of the University of Duisburg-Essen. For weak scaling, we can show very good results up to 4,096 cores. We can also present very good strong scaling results up to 864 cores