Parallel numerical methods for partial differential equations
Abstract
It has been rightly predicted that parallel computing is inevitable. This thesis attempts to study and implement the so-called geometry splitting solution paradigm as a parallel computational framework for solving elliptic partial differential equations (PDEs) on distributed memory machines. First, we formulate, analyze and implement the Schwarz Alternating Method (SAM) for elliptic PDEs defined in one and multi-dimensional domains. Specifically, we analyze SAM methods whose convergence is controlled by a different parameter in each interface condition or overlapped domain. We derive both analytical and experimental results. Furthermore, we introduce a symmetric version of SAM and make useful observations about its convergence. Second, we implement four non-overlapping geometry splitting approaches based on finite element and difference techniques. One of them is formulated on the extended rectangular domain that encapsulates the given PDE domain and its corresponding grid. This encapsulation method assumes an extension of the PDE problem outside the specified domain of definition. This approach has reduced significantly the grid partitioning overhead without reducing the overall efficiency of the computation. Finally, we have parallelized the well-known ITPACK library and implemented it on the nCUBE II machines. All discretization and solution modules developed in this thesis have been integrated in the parallel ELLPACK environment and their performance has been extensively studied.
Degree
Ph.D.
Advisors
Houstis, Purdue University.
Subject Area
Mathematics|Mechanics|Computer science
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