The analysis of iterative elliptic PDE solvers based on the cubic Hermite collocation discretization
Abstract
Collocation methods based on bicubic Hermite piecewise polynomials have been proven effective techniques for solving second-order linear elliptic PDEs with mixed boundary conditions. The corresponding linear system is in general non-symmetric and non-diagonally dominant. Iterative methods for their solution are not known and they are currently solved using Gauss elimination with scaling and partial pivoting. Point iterative methods do not converge even for the collocation equations obtained from model PDE problems. The development of efficient iterative solvers for these equations is necessary for three-dimensional problems and their parallel solution, since direct solvers tend to be space bound and their parallelization is difficult. In this thesis, we develop block iterative methods for the collocation equations of elliptic PDEs defined on a rectangle and subject to uncoupled mixed boundary conditions. For model problems of this type, we derive analytic expressions for the eigenvalues of the block Jacobi iteration matrix and determine the optimal parameter for the block scSOR method. For the case of general domains, the iterative solution of the collocation equations is still an open problem. We address this open problem by generalizing interior collocation method for PDEs defined on rectilinear regions, study the structure of these equations under different ordering schemes, and apply scAOR and scCG type iterative solvers to them. Another objective of this thesis is to study the applicability and effectiveness of geometry splitting methods coupled with collocation discretization schemes. Specifically, we consider the Generalized Schwarz Splitting (GSS) method, which is an extension of the Schwarz Alternating Method, for solving elliptic PDE problems with generalized interface conditions. The main focus is the iterative solution of the corresponding enhanced GSS linear system for a model problem. For this we carry out the spectral analysis of the enhanced block Jacobi iteration matrix. In the case of one-dimensional problems, we determine the convergence interval of one-parameter GSS and find a subinterval of it where the optimal parameter lies; moreover, we obtain sets of optimal parameters for the multi-parameter GSS case. We also analyze the convergence properties of the one-parameter GSS for a two-dimensional model problem.
Degree
Ph.D.
Advisors
Houstis, Purdue University.
Subject Area
Mathematics
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