Optimization in solving elliptic problems

✍ Scribed by Eugene G. Dyakonov

• Publisher: CRC Press
• Year: 2018
• Tongue: English
• Leaves: 591
• Series: CRC revivals
• Category: Library

✦ Table of Contents

Content: 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

✦ Subjects

Differential equations, Elliptic -- Asymptotic theory;MATHEMATICS / Calculus;MATHEMATICS / Mathematical Analysis