Thesis (Ph.D)--Washington State University, August 2006. / Includes bibliographical references (p. 91-93).

Accelerating the convergence rate of Krylov subspace methods with parallelizable preconditioners is essential for obtaining effective iterative solvers for very large linear systems of equations. Substructuring provides a framework for constructing robust and parallel preconditioners for linear systems arising from the discretization of boundary value problems. Although collocation is a very general and effective discretization technique for many PDE problems, there has been relatively little work on preconditioners for collocation problems. This thesis proposes two preconditioning methods for solving linear systems of equations arising from Hermite bicubic collocation discretization of elliptic partial differential equations on square domains with mixed boundary conditions. The first method, called <i>edge preconditioning</i>, is based on a decomposition of the domain in parallel strips, and the second, called <i>edge-vertex preconditioning</i>, is based on a two-dimensional decomposition. The preconditioners are derived in terms of two special rectangular grids -- a coarse grid with diameter <i>H</i> and a hybrid coarse/fine grid -- which together with the fine grid of diameter <i>h</i> provide the framework for approximating the interface problem induced by substructuring. We show that the proposed methods are effective for nonsymmetric indefinite problems, both from the point of view of the cost per iteration and of the number of iterations. For an appropriate choice of <i>H</i>, the edge preconditioner requires <i>O(N)</i> arithmetic operations per iteration, while the edge-vertex preconditioner requires <i>O(N^4/3)</i> operations, where <i>N</i> is the number of unknowns. For the edge-vertex preconditioner, the number of iterations is almost constant when <i>h</i> and <i>H</i> decrease such that <i>H/h</i> is held constant and it increases very slowly with <i>H</i> when <i>h</i> is held constant. For both the edge- and edge-vertex preconditioners the number of iterations depends only weakly on <i>h</i> when <i>H</i> is constant. The edge-vertex preconditioner outperforms the edge-preconditioner for small enough <i>H</i>. Numerical experiments illustrate the parallel efficiency of the preconditioners which is similar or even better than that provided by the well-known PETSc parallel software library for scientific computing. / Ph. D.

P-version finite element method for the second order elliptic equation in an arbitrary sufficiently smooth domain is studied in the frame of DD method. Two types square reference elements are used with the products of the integrated Legendre's polynomials for the coordinate functions. There are considered the estimates for the condition numbers, preconditioning of the problems arising on subdomains and the Schur complement, the derivation of the DD preconditioner. For the result we obtain the DD preconditioner to which corresponds the generalized condition number of order (logp )2 . The paper consists of two parts. In part I there are given some preliminary results for 1D case, condition number estimates and some inequalities for 2D reference element. Part II is devoted to the derivation of the Schur complement preconditioner and conditionality number estimates for the p-version finite element matrixes. Also DD preconditioning is considered.

A parallel realisation of the Conjugate Gradient Method with Schur-Complement preconditioning, based on a domain decomposition approach, is described in detail. Special kinds of solvers for the resulting interiour and coupling systems are presented. A large range of numerical results is used to demonstrate the properties and behaviour of this solvers in practical situations.

This paper concerns the solution of plate bending problems in domains composed of rectangles. Domain decomposition (DD) is the basic tool used for both the parallelization of the conjugate gradient method and the construction of efficient parallel preconditioners. A so-called Dirich- let DD preconditioner for systems of linear equations arising from the fi- nite element approximation by non-conforming Adini elements is derived. It is based on the non-overlapping DD, a multilevel preconditioner for the Schur-complement and a fast, almost direct solution method for the Dirichlet problem in rectangular domains based on fast Fourier transform. Making use of Xu's theory of the auxiliary space method we construct an optimal preconditioner for plate problems discretized by conforming Bogner-Fox-Schmidt rectangles. Results of numerical experiments carried out on a multiprocessor sys- tem are given. For the test problems considered the number of iterations is bounded independent of the mesh sizes and independent of the number of subdomains. The resulting parallel preconditioned conjugate gradient method requiresO(h^-2 ln h^-1 ln epsilon^-11) arithmetical operations per processor in order to solve the finite element equations with the relative accuracy epsilon.

In this work, prestressed elasticity problem as a model of the so-called glacial isostatic adjustment (GIA) process is studied. The model problem is described by a set of partial differential equations (PDE) and discretized with a mixed finite element (FE) formulation. In the presence of prestress the so-constructed system of equations is non-symmetric and indefinite. Moreover, the resulting system of equations is of the saddle point form. We focus on a robust and efficient block lower-triangular preconditioning method, where the lower diagonal block is and approximation of the so-called Schur complement. The Schur complement is approximated by the so-called element-wise Schur complement. The element-wise Schur complement is constructed by assembling exact local Schur complements on the cell elements and distributing the resulting local matrices to the global preconditioner matrix. We analyse the properties of the element-wise Schur complement for the symmetric indefinite system matrix and provide proof of its quality. We show that the spectral radius of the element-wise Schur complement is bounded by the exact Schur complement and that the quality of the approximation is not affected by the domain shape. The diagonal blocks of the lower-triangular preconditioner are combined with inner iterative schemes accelerated by (numerically) optimal and robust algebraic multigrid (AMG) preconditioner. We observe that on distributed memory systems, the top pivot block of the preconditioner is not scaling satisfactorily. The implementation of the methods is further studied using a general profiling tool, designed for clusters. For nonsymmetric matrices we use the theory of Generalized Locally Toeplitz (GLT) matrices and show the spectral behavior of the element-wise Schur complement, compared to the exact Schur complement. Moreover, we use the properties of the GLT matrices to construct a more efficient AMG preconditioner. Numerical experiments show that the so-constructed methods are robust and optimal.

P-version finite element method for the second order elliptic equation in an arbitrary sufficiently smooth domain is studied in the frame of DD method. Two types square reference elements are used with the products of the integrated Legendre's polynomials for the coordinate functions. There are considered the estimates for the condition numbers, preconditioning of the problems arising on subdomains and the Schur complement, the derivation of the DD preconditioner. For the result we obtain the DD preconditioner to which corresponds the generalized condition number of order (logp )2 . The paper consists of two parts. In part I there are given some preliminary results for 1D case, condition number estimates and some inequalities for 2D reference element. Part II is devoted to the derivation of the Schur complement preconditioner and conditionality number estimates for the p-version finite element matrixes. Also DD preconditioning is considered.

A parallel realisation of the Conjugate Gradient Method with Schur-Complement preconditioning, based on a domain decomposition approach, is described in detail. Special kinds of solvers for the resulting interiour and coupling systems are presented. A large range of numerical results is used to demonstrate the properties and behaviour of this solvers in practical situations.

This paper concerns the solution of plate bending problems in domains composed of rectangles. Domain decomposition (DD) is the basic tool used for both the parallelization of the conjugate gradient method and the construction of efficient parallel preconditioners. A so-called Dirich- let DD preconditioner for systems of linear equations arising from the fi- nite element approximation by non-conforming Adini elements is derived. It is based on the non-overlapping DD, a multilevel preconditioner for the Schur-complement and a fast, almost direct solution method for the Dirichlet problem in rectangular domains based on fast Fourier transform. Making use of Xu's theory of the auxiliary space method we construct an optimal preconditioner for plate problems discretized by conforming Bogner-Fox-Schmidt rectangles. Results of numerical experiments carried out on a multiprocessor sys- tem are given. For the test problems considered the number of iterations is bounded independent of the mesh sizes and independent of the number of subdomains. The resulting parallel preconditioned conjugate gradient method requiresO(h^-2 ln h^-1 ln epsilon^-11) arithmetical operations per processor in order to solve the finite element equations with the relative accuracy epsilon.

Este trabalho apresenta a aplicação de técnicas de Decomposição de Domínio e Processamento Paralelo na solução de grandes sistemas de equações algébricas lineares provenientes da modelagem de fenômenos eletromagnéticos pelo Método de Elementos Finitos. Foram implementadas as técnicas dos tipos Complemento de Schur e o Método Aditivo de Schwarz, adaptadas para a resolução desses sistemas em cluster de computadores do tipo Beowulf e com troca de mensagens através da Biblioteca MPI. A divisão e balanceamento de carga entre os processadores são feitos pelo pacote METIS. Essa metodologia foi testada acoplada a métodos, seja iterativo (ICCG), seja direto (LU) na etapa de resolução dos sistemas referentes aos nós internos de cada partição. Para a resolução do sistema envolvendo os nós de fronteira, no caso do Complemento de Schur, utilizou-se uma implementação paralisada do Método de Gradientes Conjugados (PCG). S~ao discutidos aspectos relacionados ao desempenho dessas técnicas quando aplicadas em sistemas de grande porte. As técnicas foram testadas na solução de problemas de aplicação do Método de Elementos Finitos na Engenharia Elétrica (Magnetostática, Eletrocinética e Magnetodinâmica), sejam eles de natureza bidimensional com malhas não estruturadas, seja tridimensional, com malhas estruturadas. / This work presents the study of Domain Decomposition and Parallel Processing Techniques applied to the solution of systems of algebraic equations issued from the Finite Element Analysis of Electromagnetic Phenomena. Both Schur Complement and Schwarz Additive techniques were implemented. They were adapted to solve the linear systems in Beowulf clusters with the use of MPI library for message exchange. The load balance among processors is made with the aid of METIS package. The methodology was tested in association to either iterative (ICCG) or direct (LU) methods in order to solve the system related to the inner nodes of each partition. In the case of Schur Complement, the solution of the system related to the boundary nodes was performed with a parallelized Conjugated Gradient Method (PCG). Some aspects of the peformance of these techniques when applied to large scale problems have also been discussed. The techniques has been tested in the simulation of a collection of problems of Electrical Engineering, modelled by the Finite Element Method, both in two dimensions with unstructured meshes (Magnetostatics) and three dimensions with structured meshes (Electrokinetics).