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Optimization in solving elliptic problems

Author: E G Dʹi︠a︡konov; S F McCormick
Publisher: Boca Raton, FL : CRC Press, [2018]
Series: CRC revivals.
Edition/Format:   eBook : Document : EnglishView all editions and formats
Summary:
Optimization in Solving Elliptic Problems focuses on one of the most interesting and challenging problems of computational mathematics - the optimization of numerical algorithms for solving elliptic problems. It presents detailed discussions of how asymptotically optimal algorithms may be applied to elliptic problems to obtain numerical solutions meeting certain specified requirements. Beginning with an outline of  Read more...
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Genre/Form: Electronic books
Additional Physical Format: Print version:
(OCoLC)1027833796
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: E G Dʹi︠a︡konov; S F McCormick
ISBN: 9781351075213 1351075217 9781351083669 135108366X 9781351092111 1351092111
OCLC Number: 1035748802
Description: 1 online resource.
Contents: Cover; Half Title; Title Page; Copyright Page; Preface; Editor's Preface; The Author; The Editor; Basic Notation; Table of Contents; Introduction; 1. Modern formulations of elliptic boundary value problems; 1.1. Variational principles of mathematical physics; 1.2. Variational problems in a Hilbert space; 1.3. Completion of a preHilbert space and basic properties of Sobolev spaces; 1.4. Generalized solutions of elliptic boundary value problems; 2. Projective-grid methods (finite element methods); 2.1. Rayleigh-Ritz method; 2.2. Bubnov-Galerkin method and projective methods 2.3. Projective-grid methods (finite element methods)2.4. The simplest projective-grid operators; 2.5. Composite grids and triangulations; local grid refinement; 3. Methods of solution of discretized problems; asymptotically optimal and nearly optimal preconditioners; 3.1. Specificity of grid systems; direct methods; 3.2. Classical iterative methods; 3.3. Iterative methods with spectrally equivalent operators; optimal preconditioning; 3.4. Symmetrizations of systems; 3.5. Coarse grid continuation (multigrid acceleration of the basic iterative algorithm) 3.6. Some nonelliptic applications 4. Invariance of operator inequalities under projective approximations; 4.1. Rayleigh-Ritz method and Gram matrices; 4.2. Projective approximations of operators; 4.3. Spectral equivalence of grid operators defined on topologically equivalent triangulations; 4.4. Spectral equivalence of grid operators defined on composite triangulations with local refinements; 5. N-widths of compact sets and optimal numerical methods for classes of problems; 5.1. Approximations of compact sets and criteria for optimality of computational algorithms 3. Iterative methods with model symmetric operators3.1. Estimates of rates of convergence in the Euclidean space H(B) of the modified method of the simple iteration; 3.2. Estimates of the rate of convergence in the Euclidean space H(B2); 3.3. Condition numbers of symmetrized linear systems; generalizations for nonlinear problems; 3.4. A posteriori estimates; 3.5. Modifications of Richardson's iteration; 3.6. Use of orthogonalization; 3.7. Adaptation of iterative parameters; 3.8. Modified gradient methods; 3.9. Nonsymmetric model operators
Series Title: CRC revivals.
Other Titles: Minimizat︠s︡ii︠a︡ vychislitelʹnoĭ raboty.
Responsibility: by Eugene G. D'yakonov ; Steve McCormick, editor of the English translation.

Abstract:

Optimization in Solving Elliptic Problems focuses on one of the most interesting and challenging problems of computational mathematics - the optimization of numerical algorithms for solving elliptic problems. It presents detailed discussions of how asymptotically optimal algorithms may be applied to elliptic problems to obtain numerical solutions meeting certain specified requirements. Beginning with an outline of the fundamental principles of numerical methods, this book describes how to construct special modifications of classical finite element methods such that for the arising grid systems, asymptotically optimal iterative methods can be applied. Optimization in Solving Elliptic Problems describes the construction of computational algorithms resulting in the required accuracy of a solution and having a pre-determined computational complexity. Construction of asymptotically optimal algorithms is demonstrated for multi-dimensional elliptic boundary value problems under general conditions. In addition, algorithms are developed for eigenvalue problems and Navier-Stokes problems. The development of these algorithms is based on detailed discussions of topics that include accuracy estimates of projective and difference methods, topologically equivalent grids and triangulations, general theorems on convergence of iterative methods, mixed finite element methods for Stokes-type problems, methods of solving fourth-order problems, and methods for solving classical elasticity problems. Furthermore, the text provides methods for managing basic iterative methods such as domain decomposition and multigrid methods. These methods, clearly developed and explained in the text, may be used to develop algorithms for solving applied elliptic problems.

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