r Chapter 12 Symbolic and Parallel Adaptive Methods for Partial Differential Equations

All Rights Reserved No part of this book may be reproduced in any form, by photostat, microfilm or any other means, without written permission from the publishers A catalogue record for this book is available from the British Library In order to enhance the use of adaptive mesh refinement and order enrichment methods for partial differential equations by practising scientists and engineers, we have developed a symbolic interface to the numerical software. Partial differential equations and data are described in a natural high-level language. FORTRAN code, required by the various numerical procedures for evaluating functions, Jacobians, etc. is automatically generated and properties of the system, such as linearity and symmetry, are deduced. Parallel procedures for adaptive techniques are of high interest given the need to solve problems of increasing complexity. We describe techniques for solving two-dimensional vector systems of elliptic and hyperbolic partial differential equations on shared-memory parallel computers. Linear algebraic systems resulting from the finite element discretiza-tion of an elliptic problem using a hierarchical piecewise polynomial basis on a finite-quadtree-structured mesh are solved by a conjugate gradient technique with a symmetric successive over-relaxation preconditioner. System assembly and solution are processed in parallel with computations scheduled on noncontiguous quadrants of the tree in order to minimize process synchronization. Coloring unstructured meshes that result from quadtrees is far simpler than coloring more general meshes, and we describe a linear time complexity coloring procedure that uses a maximum of six colors. Hyperbolic systems are approximated by an explicit finite volume technique and solved by a recursive local mesh refinement procedure on a tree-structured grid. Computational procedures that sequentially traverse the tree while processing solutions on each grid in parallel, that process solutions at the same tree level in parallel, and that dynamically assign processors to nodes of the tree have been developed and applied to an example that illustrates their performance.

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