Adaptive Grid Conference Edinburgh 96: flaherty Abstract

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Parallel adaptive techniques for partial differential equations

J E Flaherty (Invited Presentation)

Department of Computer Science
and
Scientific Computation Research Center
Rensselaer Polytechnic Institute
Troy, New York 12180-3590 USA

Abstract

Adaptive computational techniques are having a dramatic impact on the way that scientists and engineers solve problems involving partial differential equations. Specific adaptive strategies enrich preliminary solutions obtained on coarse meshes using combinations of mesh refinement and coarsening (h-refinement), variation of method order (p-refinement), and mesh motion (r-refinement). A posteriori estimates of discretization errors control adaptive enrichment and provide a measure of solution reliability.

Like adaptivity, parallel computation is being used to an increasing degree as multi-dimensional, nonlinear, and other realistic features are included in mathematical models. However, adaptivity and parallelism are at odds. The most efficient adaptive procedures utilize hp-refinement to achieve methods having very high convergence rates. Optimality occurs through complex logic and heuristics that are difficult to parallelize.

We have developed a framework for adaptive and parallel computation that is capable of (i) generating three-dimensional unstructured meshes of tetrahedral elements, (ii) automatically refining and coarsening these meshes, (iii) partitioning the computation into subdomains that may be processed in parallel, and (iv) maintaining a balanced parallel computation through element migration. Mesh generation is supported by a hierarchical database in which spatial regions are linked to their bounding faces, which are linked to their bounding edges, which are linked to their endpoints. The database is connected to geometrical modeling procedures and to a parallel library having capabilities for processor scheduling and reassignment.

The adaptive finite element framework is being used to study several flow problems associated with materials fabrication. These include the fabrication of ceramic composites, the coating of ceramic fibers, the oxidation of ceramic composites, and the manufacture of crystals. The adaptive software is capable of following the reaction fronts and phase boundaries associated with these problems and accounting for the material distortion that accompanies some reactions.

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Last modified Fri Jun 21 19:19:13 GB-Eire 1996 (DBD)