Parallel solution of unstructured sparse finite element equations

3D EM simulation is computationally demanding. A finite element partial differential equation solution of a scattering problem requires the solution of an equation of the form Ax=b, where A is very large, very sparse, is unstructured, and may have complex elements and be non-symmetric. With a cartesian grid, natural node numbering produces a multi-diagonal A-matrix. Using triangular (or tetrahedral in 3D) elements results in an unstructured matrix. Node renumbering schemes may be used to produce a banded matrix. The non-zero elements only, together with a pointer, can be stored to represent the A-matrix. When the matrix size exceeds the collective memory available, an iterative solver becomes essential. Single instruction multiple data (SIMD) parallel machines represent a cost-effective platform, but their single instruction stream makes load balancing difficult, as compared with multiple instruction (MIMD) parallel machines. Conclusions from a SIMD implementation should be useful for MIMD machines. The conjugate gradient (CG) algorithm and its variants, in particular, the CG squared algorithm (CGS), were selected for initial study [Nachtigal et al., 1992]. Purdue's MasPar 16K processor MP-1 machine was used for the implementation of a new matrix vector multiplication scheme employed repeatedly in the iterative solution. An example 2D scattering problem, using a variational formulation and having a modest number of unknowns, was considered. The complex sparse system was solved using the CGS iterative solver, which involves a series of sparse matrix-vector multiplication operations.