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Preface
Editor's Preface
The Author
The Editor
Basic Notation
Introduction
Modern formulations of elliptic boundary value problems
Variational principles of mathematical physics
Variational problems in a Hilbert space
Completion of a preHilbert space and basic properties of Sobolev spaces
Generalized solutions of elliptic boundary value problems
Projective-grid methods (finite element methods)
Rayleigh-Ritz method
Bubnov-Galerkin method and projective methods
Projective-grid methods (finite element methods)2.4. The simplest projective-grid operators
Composite grids and triangulations
local grid refinement
Methods of solution of discretized problems
asymptotically optimal and nearly optimal preconditioners
Specificity of grid systems
direct methods
Classical iterative methods
Iterative methods with spectrally equivalent operators
optimal preconditioning
Symmetrizations of systems
Coarse grid continuation (multigrid acceleration of the basic iterative algorithm)
Some nonelliptic applications
Invariance of operator inequalities under projective approximations
Rayleigh-Ritz method and Gram matrices
Projective approximations of operators
Spectral equivalence of grid operators defined on topologically equivalent triangulations
Spectral equivalence of grid operators defined on composite triangulations with local refinements
N-widths of compact sets and optimal numerical methods for classes of problems
Approximations of compact sets and criteria for optimality of computational algorithms
Iterative methods with model symmetric operators
Estimates of rates of convergence in the Euclidean space H(B) of the modified method of the simple iteration
Estimates of the rate of convergence in the Euclidean space H(B2)
Condition numbers of symmetrized linear systems
generalizations for nonlinear problems
A posteriori estimates
Modifications of Richardson's iteration
Use of orthogonalization
Adaptation of iterative parameters
Modified gradient methods
Nonsymmetric model operators