Thesis (Ph.D.)--Mississippi State University. Department of Mathematics and Statistics. / Title from title screen. Includes bibliographical references.

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Enclosure theorems for the eigenvalues and representational formulae for the eigenfunctions of a linear, elliptic, second order partial differential operator will be established for specific domain perturbations to which the classical theory cannot be applied. In particular, the perturbation of n-dimensional Euclidean space E[superscript]n to an n-disk D[subscript]a of radius a is considered in Chapter I and the perturbation of the upper half-space H[superscript]n of E[superscript]n to the upper half of D[subscript]a, S[subscript]a, is discussed in Chapter II. In each case a general self-adjoint boundary condition is adjoined on the bounding surface of the perturbed domain. / Science, Faculty of / Mathematics, Department of / Graduate

This thesis studies the existence, multiplicity, bifurcation and the stability of the solutions to semilinear elliptic boundary value problems. These problems are motivated both by the mathematical structure and the numerous applications in fluid mechanics chemical reactions, nuclear reactors, Riemannian geometry and elasticity theory. This study considers the problem for different classes of nonlinearities and obtain the existence and multiplicity of positive solutions.

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Numerical solutions of partial differential equations (PDEs) resulting from problems in both the engineering and natural sciences result in solving large sparse linear systems Au = b. The construction of such linear systems and their solutions using either direct or iterative methods are topics of continuing research. The recent advent of parallel computer architectures has resulted in a search for good parallel algorithms to solve such systems, which in turn has led to a recent burgeoning of research into domain decomposition algorithms. Domain decomposition is a procedure which employs subdivision of the solution domain into smaller regions of convenient size or shape and, although such partitionings have proven to be quite effective on serial computers, they have proven to be even more effective on parallel computers. Recent work in domain decomposition algorithms has largely been based on second order accurate discretization techniques. This dissertation describes an algorithm for the numerical solution of general two-dimensional linear elliptic partial differential equations with variable coefficients which employs both a high order accurate discretization and a Krylov subspace iterative solver in which a preconditioner is developed using domain decomposition. Most current research into such algorithms has been based on symmetric systems; however, variable PDE coefficients generally result in a nonsymmetric A, and less is known about the use of preconditioned Krylov subspace iterative methods for the solution of nonsymmetric systems. The use of the high order accurate discretization together with a domain decomposition based preconditioner results in an iterative technique with both high accuracy and rapid convergence. Supporting theory for both the discretization and the preconditioned iterative solver is presented. Numerical results are given on a set of test problems of varying complexity demonstrating the robustness of the algorithm. It is shown that, if only second order accuracy is required, the algorithm becomes an extremely fast direct solver. Parallel performance of the algorithm is illustrated with results from a shared memory multiprocessor. / Ph. D.

In general relativity, angular momentum of the gravitational field in some volume bounded by an axially symmetric sphere is well-defined as a boundary integral. The definition relies on the symmetry generating vector field, a Killing field, of the boundary. When no such symmetry exists, one defines angular momentum using an approximate Killing field. Contained in the literature are various approximations that capture certain properties of metric preserving vector fields. We explore the continuity of an angular momentum definition that employs an approximate Killing field that is an eigenvector of a particular second-order differential operator. We find that the eigenvector varies continuously in Hilbert space under smooth perturbations of a smooth boundary geometry. Furthermore, we find that not only is the approximate Killing field continuous but that the eigenvalue problem which defines it is stable in the sense that all of its eigenvalues and eigenvectors are continuous in Hilbert space. We conclude that the stability follows because the eigenvalue problem is strongly elliptic. Additionally, we provide a practical introduction to the mathematical theory of strongly elliptic operators and generalize the above stability results for a large class of such operators. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2014. / FAU Electronic Theses and Dissertations Collection

Li Kin-kuen. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (leaves 52-54). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.4 / Chapter 2 --- Variational Formulation --- p.10 / Chapter 3 --- The Existence Of A Mountain Pass Solution --- p.12 / Chapter 4 --- Ground States --- p.21 / Chapter 5 --- The Projections Of v And w --- p.35 / Chapter 6 --- Computation Of The Energy: An Upper Bound --- p.37 / Chapter 7 --- Convergence: The First Approximation --- p.40 / Chapter 8 --- Convergence: The Second Approximation --- p.44 / Chapter 9 --- Computation Of The Energy: A Lower Bound --- p.48 / Chapter 10 --- Comparing The Energy: Completion Of The Proof Of Theorem 1.2 --- p.51 / Bibliography --- p.52

We study Monge-Ampère-type equations on compact complex manifolds. We prove a C² estimate for solutions to a general class of non-concave parabolic equations, extending work from the Kähler setting. Next we prove C⁰, C², and curvature estimates for solutions to a particular continuity path of elliptic equations on specific examples of non-Kähler manifolds, adapting work on the Chern-Ricci flow. In each case the estimates give a certain type of convergence of the solutions. The estimates are obtained by maximum principle arguments, and in the first part of this work we set up a general framework that facilitates the various C² estimates which follow.

Orientador: Sônia Maria Gomes, Philippe Remy Bernard Devloo, Igor Mozolevski / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-19T03:23:02Z (GMT). No. of bitstreams: 1 Goncalves_JoaoLuis_D.pdf: 15054031 bytes, checksum: 23ef9ef75ca5a7ae7455135fc552a678 (MD5) Previous issue date: 2011 / Resumo: Neste trabalho, estudamos indicadores a posteriori para o erro na aproximação de funcionais das soluções das equações biharmônica e de Poisson obtidas pelo método de Galerkin descontínuo. A metodologia usada na obtenção dos indicadores é baseada no problema dual associado ao funcional, que é conhecida por gerar os indicadores mais eficazes. Os dois principais indicadores de erro com base no problema dual já obtidos, apresentados para problemas de segunda ordem, são estendidos neste trabalho para problemas de quarta ordem. Também propomos um terceiro indicador para problemas de segunda e quarta ordem. Estudamos as características dos diferentes indicadores na localização dos elementos com as maiores contribuições do erro, na caracterização da regularidade das soluções, bem como suas consequências na eficiência dos indicadores. Estabelecemos uma estratégia hp-adaptativa específica para os indicadores de erro em funcionais. Os experimentos numéricos realizados mostram que a estratégia hp-adaptativa funciona adequadamente e que o uso de espaços de aproximação hp-adaptados resulta ser eficiente para a redução do erro em funcionais com menor úmero de graus de liberdade. Além disso, nos exemplos estudados, a qualidade dos resultados varia entre os indicadores, dependendo do tipo de singularidade e da equação tratada, mostrando a importância de dispormos de uma maior diversidade de indicadores / Abstract: In this work we study goal-oriented a posteriori error indicators for approximations by the discontinuous Galerkin method for the biharmonic and Poisson equations. The methodology used for the indicators is based on the dual problem associated with the functional, which is known to generate the most effective indicators. The two main error indicators based on the dual problem, obtained for second order problems, are extended to fourth order problems. We also propose a third indicator for second and fourth order problems. The characteristics of the different indicators are studied for the localization of the elements with the greatest contributions of the error, and for the characterization of the regularity of the solutions, as well as their consequences on indicators efficiency. We propose an hp-adaptive strategy specific for goal-oriented error indicators. The performed numerical experiments show that the hp-adaptive strategy works properly, and that the use of hp-adapted approximation spaces turns out to be efficient to reduce the error with a lower number of degrees of freedom. Moreover, in the examples studied, a comparison of the quality of results for the different indicators shows that it may depend on the type of singularity and of the equation treated, showing the importance of having a wider range of indicators / Doutorado / Matematica Aplicada / Doutor em Matemática Aplicada