Date of Award

Spring 2010

Document Type

Thesis

Degree Name

Doctor of Philosophy (PhD)

Department

Applied Mathematics

First Advisor

Thomas A. Manteuffel

Second Advisor

Stephen F. McCormick

Third Advisor

Marian Brezina

Abstract

New adaptive local refinement (ALR) strategies are developed, the goal of which is to reach a given error tolerance with the least amount of computational cost. This strategy is especially attractive in the setting of a first-order system least-squares (FOSLS) finite element formulation in conjunction with algebraic multigrid (AMG) methods in the context of nested iteration (NI). To accomplish this, the refinement decisions are determined based on minimizing the predicted `accuracy-per-computational-cost' efficiency (ACE). The nested iteration approach produces a sequence of refinement levels in which the error is equally distributed across elements on a relatively coarse grid. Once the solution is numerically resolved, refinement becomes nearly uniform. Efficiency of the algorithms are demonstrated through a 2D Poisson problem with steep gradients, and 2D reduced model of the incompressible, resistive magnetohydrodynamic (MHD) equations.

Accommodations of the ALR strategies to parallel computer architectures involve a geometric binning strategy to reduce communication cost. Load balancing begins at very coarse levels. Elements and nodes are redistributed using parallel quad-tree structures and a space filling curve. This automatically ameliorates load balancing issues at finer levels. Numerical results produced on Frost, the NCAR/CU Blue Gene/L supercomputer, are presented for a 2D Poisson problem with steep gradients, a 2D backward facing step incompressible Stokes equations and Navier-Stokes equations. The NI-FOSL-AMG-ACE approach is able to provide highly resolved approximations to rapidly varying solutions using a small number of work units. Excellent weak and strong scalability of parallel ALR are demonstrated on up to 4,096 processors for problems with up to 15 million biquadratic elements.

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