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Techniques for the adaptive solution of two-dimensional vector systems for hyperbolic and elliptic partial differential equations on shared-memory parallel computers are described. Hyperbolic systems are approximated by an explicit finite volume technique and solved by a recursive local mesh refinement procedure. Several computational procedures have been developed, and results comparing a variety of heuristic processor load-balancing techniques and refinement strategies are presented. For elliptic problems, the spatial domain is discretized using a finite quadtree mesh-generation procedure and the differential system is discretized by a finite-element Galerkin technique with a hierarchical piecewise polynomial basis. Resulting linear algebraic systems are solved in parallel on noncontiguous quadrants by a conjugate gradient technique with element-by-element and symmetric successive over-relaxation preconditioners. Noncontiguous regions are determined by using a linear-time complexity coloring procedure that requires a maximum of six colors.
Date of Publication: Sep 1991