Preconditioning and the Conjugate Gradient Method in the Context of Solving PDEs

Abstract

Preface 1. Introduction 2. Linear elliptic partial differential equations 3. Elements of functional analysis 4. Riesz map and operator preconditioning 5. Conjugate gradient method in Hilbert spaces 6. Finite dimensional Hilbert spaces and the matrix formulation of the conjugate gradient method 7. Comments on the Galerkin discretization 8. Preconditioning of the algebraic system as transformation of the discretization basis 9. Fundamental theorem on discretization 10. Local and global information in discretization and in computation 11. Limits of the condition number-based descriptions 12. Inexact computations, a posteriori error analysis, and stopping criteria 13. Summary and outlook Bibliography Index.