Reordering and Balanced Mesh Partitioning for the Schur Complement Method

and contributions A finite element method often leads to large sparse symmetric and positive definite systems of linear equations. If memory capacity or computing performance of a single CPU are not sufficient for solving such large linear systems, then parallelisation must be used. The parallel solution by the Schur complement method, on homogeneous parallel machines with distributed memory, is considered. A finite element mesh is partitioned by graph partitioning. Such partitioning results in submeshes with similar numbers of elements. Consequently, mesh partitioning yields domain decomposition in submatrices of similar sizes. The submatrices are partially factorised to compute Schur complements. Prior to the solution, the variables are reordered to minimise the memory requirements to store the submatrices and to minimise the time of the partial factorisation. However, the reordering algorithms designed for the sequential solution are commonly used in parallel solver. The first contribution of this thesis are two improved reordering algorithms for the needs of partial factorisation. Classic graph partitioning produces submatrices of similar sizes. However, the memory requirements to store them, or the time spent on partial factorisation can be different, i.e., disbalanced. This is because methods exploiting the sparsity of submatrices are used. A memory disbalanced requirement may cause the crash of the solver and the disbalanced partial factorisation times prolong the overall computation time. The second contribution of this thesis is a Quality Balancing heuristic that modifies the classic mesh partitioning to balance the given quality, i.e., memory requirements to store submatrices or the partial factorisation times.