Numerical Methods for Grid Equations Volume II Iterative Methods [Paperback]

5 The Mathematical Theory of Iterative Methods.- 5.1 Several results from functional analysis.- 5.1.1 Linear spaces.- 5.1.2 Operators in linear normed spaces.- 5.1.3 Operators in a Hilbert space.- 5.1.4 Functions of a bounded operator.- 5.1.5 Operators in a finite-dimensional space.- 5.1.6 The solubility of operator equations.- 5.2 Difference schemes as operator equations.- 5.2.1 Examples of grid-function spaces.- 5.2.2 Several difference identities.- 5.2.3 Bounds for the simplest difference operators.- 5.2.4 Lower bounds for certain difference operators.- 5.2.5 Upper bounds for difference operators.- 5.2.6 Difference schemes as operator equations in abstract spaces.- 5.2.7 Difference schemes for elliptic equations with constant coefficients.- 5.2.8 Equations with variable coefficients and with mixed derivatives.- 5.3 Basic concepts from the theory of iterative methods.- 5.3.1 The steady state method.- 5.3.2 Iterative schemes.- 5.3.3 Convergence and iteration counts.- 5.3.4 Classification of iterative methods.- 6 Two-Level Iterative Methods.- 6.1 Choosing the iterative parameters.- 6.1.1 The initial family of iterative schemes.- 6.1.2 The problem for the error.- 6.1.3 The self-adjoint case.- 6.2 The Chebyshev two-level method.- 6.2.1 Construction of the set of iterative parameters.- 6.2.2 On the optimality of the a priori estimate.- 6.2.3 Sample choices for the operator D.- 6.2.4 On the computational stability of the method.- 6.2.5 Construction of the optimal sequence of iterative parameters.- 6.3 The simple iteration method.- 6.3.1 The choice of the iterative parameter.- 6.3.2 An estimate for the norm of the transformation operator.- 6.4 The non-self-adjoint case. The simple iteration method.- 6.4.1 Statement of the problem.- 6.4.2 Minimizing the norm of the transformation operator.- 6.4.3 Minimizing the norm of the resolving operator.- 6.4.4 The symmetrization method.- 6.5 Sample applications of the iterative methods.- 6.5.1 A Dirichlet difference problem for Poisl#-