Heuristics for Domain Decomposition

The domain decomposition method embodies large potentials for a parallelization of FEM methods. In this data-parallel approach, the domain of interest is partitioned into smaller subdomains, either before the mesh generation or afterwards (mesh partitioning). The subdomains are assigned to processors that calculate the corresponding part of the approximation. The mesh decomposition and assignments of subdomains to processors can be modeled as graph embedding problem where a large graph (the mesh) has to be mapped onto a smaller one (the processor network). Unfortunately, this mapping problem is NP-complete and there exist almost no eecient sequential or parallel heuristics that solve this problem suuciently 1]. With growing performance of interconnection networks and especially with the establishment of independent routing networks, it is appropriate to reduce the mapping problem to the task of partitioning the graph (the FEM-mesh) into as many equal sized (or weighted) clusters as there are numbers of processors and to minimize the number of edges crossing the partition boundaries. For the special application of partitioning FEM-meshes heuristics have to be exible with respect to the measures they optimize. Graph partitioning in general is a combinatorial optimization problem. Normal partitioning heuristics minimize the total cut size, i.e. the number of edges crossing partition boundaries. This makes sense in order to reduce the amount of necessary communication of the parallel FEM-simulation. But there are several other measures that are often much more important than cut size and that depend very much on the numerical solution method used for the simulation 4]. Examples are the aspect ratio of subdomains or their convexity. A large number of eecient graph partitioning heuristics have been designed in the past, most of them for recursive bisection but also some for direct k-partitioning. See 2, 3, 6] for overviews. Many of the most eecient methods have been collected into the Chaco library by Hendrickson and Leland 7]. Farhat describes in 4] a simple and eecient sequential algorithm for the partitioning of FEM-meshes. This front-technique is a breath-rst-search based method which is widely used by engineers and innuenced the development of a number of other partitioning tools (e.g. 9]). Recursive Orthogonal Bisection (ROB) and Unbalanced Recursive Bisection (URB) use node-coordinates to partition a mesh and neglect the graph structure 8]. Both methods are fast and easy to implement whereas URB ooers larger exibility. Walshaw and Berzins apply Recursive Spectral Bisection (RSB) which was introduced by Simon et. …