A Study of the Factorization Fill-In for a Parallel Implementation of the Finite Element Method

In this paper we investigate the additional storage overhead needed for a parallel implementation of nite element applications. In particular, we compare the storage requirements for the factorization of the sparse matrices that would occur on parallel processor versus a uniprocessor. This variation in storage results from the factorization ll-in. We address the question of whether the storage overhead is so large for parallel implementations that it imposes severe limitations on the problem size in contrast to the problems executed sequentially on a uniprocessor. The storage requirements for the parallel implementation is based upon a new ordering scheme, the Combination Mesh-Based scheme. This scheme uses a domain decomposition method which attempts to balance the processors' loads and decrease the interpro-cessor communication. The storage requirements for the sequential implementation is based upon the Minimum Degree algorithm. The diierence between the two storage requirements corresponds to the storage overhead attributed to the parallel scheme. Experiments were conducted on regular and irregular, 2-D and 3-D problems. The meshes were decomposed into two through 256 subdomains which can be executed on two through 256 processors, respectively. The total storage requirements or ll-in for most of the 2-D problems were less than a factor of two increase over the 0 1 sequential execution. In contrast, large 3-D problems had zero increase in storage or ll-in over the sequential execution; the ll-in was less for the parallel execution than the sequential execution. Thus, we conclude that the storage overhead attributed to the use of parallel processors will not impose severe contraints on the problem size. Further, for large 3-D applications, the Combination Mesh-Based algorithm does better than Minimum Degree for reducing the ll-in.