Parallelization Strategies for Matrix Assembly in Finite Element Methods

Abstract

Adaptive finite element methods include unstructured discretization meshes. Most algorithms for the matrix assembly follow the ordering of the arising meshes. Time-dependent problems resulting in rapidly changing locally refined (i.e. unstructured) discretization grids need a matrix assembly in each timestep. This demands a considerable amount of computing time which in turn calls for parallelization.

Domain decomposition techniques often suffer from insufficient load balancing. Especially in adaptive methods balancing the computational workload is difficult, because the number of unknowns in each domain is not predictable and even may change during the computation.

Analysing data dependencies in the matrix assembly, one obtains the problem to synchronize access to one node (matrix element) from neighbouring elements. An efficient algorithm based on coloring the elements following a very simple tabulated indexing scheme has been found. Within each color, calculation is free of data dependencies. Load balancing becomes easy, because data can be distributed uniformly among the processors. This method works well for shared memory architectures.

For distributed memory computers, however, the color indexing method fails, because logical and physical ordering differ, thus physical data access is not structured. A second method for parallelizing the same problem (i.e. matrix assembly in finite element methods) has been developed. Using index-arrays for mapping the logical ordering to a suitable physical (storage-) ordering, one obtains an almost optimally load balanced and efficient algorithm.

Timing results for an Alliant FX/2800 (color indexing) and a Kendall Square Research KSR-1 show the efficiency of both algorithms.

Timing results for an Alliant FX/2800 (color indexing) and a Kendall Square Research KSR-1 show the efficiency of both algorithms.