Parallel and adaptive algorithms for elliptic partial differential systems

Miscellaneous · Wed Jan 01 00:00:00 EST 1992

OSTI ID:5569666

Benantar, M

The author considers the solution of two-dimensional vector systems of quasilinear elliptic partial differential equations on a shared memory parallel computer. The spatial domain is discretized using finite quadtree mesh generation procedures and the differential system is discretized by a finite element-Galerkin technique with a piecewise polynomial basis. Both stiffness matrix generation and linear system solution are processed in parallel. Resulting linear algebraic systems are solved using the conjugate gradient technique with element-by-element and symmetric over-relaxation preconditioners. Parallel computations are scheduled on noncontinguous regions of the domain in order to minimize process synchronization. To partition the spatial domain, he takes advantage of the regular quadtree structure and colors the leaf nodes of the quadtree, called quadrants, instead of elements of the unstructured grid that it generates. He describes linear-time complexity coloring procedures that use six and eight colors. In the six-color algorithm he introduces the quasi-binary tree representation of the quadtree that is used to formally prove the correctness of the coloring process. Based on the solutions of model elliptic problems, the six-color procedure provides superior parallel performance due to its increased data granularity. Higher-order approximations, typically, use coarse grids. With the number of quadrants remaining fixed and relatively small, the parallel computing approach that uses quadrants to schedule tasks results in poor performance for a p-refinement adaptive method. To remedy this problem, triangle graphs are introduced and their three colorability is applied to triangular grids in order to achieve finer data granularity by using edges of the elements to schedule parallel tasks instead of quandrants. Higher order solutions of the model problems show a superior performance of this approach than obtained using quadrants for parallel computation.

• Research Organization: Rensselaer Polytechnic Inst., Troy, NY (United States)
• OSTI ID: 5569666
• Resource Relation: Other Information: Thesis (Ph.D.)
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE
PARTIAL DIFFERENTIAL EQUATIONS
PARALLEL PROCESSING
ALGORITHMS
GALERKIN-PETROV METHOD
MATHEMATICS
MESH GENERATION
CALCULATION METHODS
DIFFERENTIAL EQUATIONS
EQUATIONS
ITERATIVE METHODS
MATHEMATICAL LOGIC
PROGRAMMING
990200* - Mathematics & Computers