An hybrid direct-iterative solver based on the Schur complement approach

An hybrid direct-iterative solver based on the Schur complement approach. The resolution of large sparse linear systems is often the most consuming step in scientific applications. Parallel sparse direct solver are now able to solve efficiently real-life three-dimensional problems having in the order of several millions of equations, but anyway they are limited by the memory requirement. On the other hand, the iterative methods require less memory, but they often fail to solve ill-conditioned systems. We propose an hybrid direct-iterative method which aims at bridging the gap between these two classes of method. The keypoint of our method is to defined an ordering and a partitioning of the unknowns that relies on a form of nested dissection ordering in which cross points in the separators play a special role. The subgraphs obtained by the nested dissection correspond to the unknowns that are eliminated using a direct method and the Schur complement system on the remaining of the unknowns (that correspond to the interface between the subdomains) is solved using an iterative method. This special ordering and partitioning allows the use of dense block algorithms both in the direct and iterative part of the solver and provides a high degree of parallelism to these algorithms. We also propose several algorithmic variants to solve the Schur complement system