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

The route from an applied problem to its numerical solution involves modeling, analysis, discretization, and the solution of the discretized problem. This book concerns the interplay of these stages and the challenges that arise. The authors link analysis of PDEs, functional analysis, and calculus of variations with iterative matrix computation using Krylov subspace methods. While preconditioning of the conjugate gradient method is traditionally developed algebraically using the preconditioned finite-dimensional algebraic system, the authors develop connections between preconditioning and PDEs. Additionally, links between the infinite-dimensional formulation of the conjugate gradient method, its discretization and preconditioning are explored. The book is intended for mathematicians, engineers, physicists, chemists, and any other researchers interested in the issues discussed. Aiming to improve understanding between researchers working on different solution stages, the book challenges commonly held views, addresses widespread misunderstandings, and formulates thought-provoking open questions for further research.