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.

Josef Málek is a Professor at the Faculty of Mathematics and Physics, Charles University in Prague, Czech Republic. He is a director of the Nečas Center for Mathematical Modeling and head of the Department of Mathematical Modeling, comprising researchers with diverse backgrounds focused upon the graduate study program 'Mathematical modeling in science and technology'. His research primarily concerns mathematical analysis of nonlinear PDEs stemming from non-Newtonian fluid mechanics, and he has contributed to the constitutive theory in the fluid and solid mechanics and in the theory of mixtures. His research approach emphasizes the need for interactions between modeling, analysis, scientific computing, and experiments.

Zdenek Strakos is a Professor at the Faculty of Mathematics and Physics, Charles University in Prague, Czech Republic. He has been an active member of several professional committees and journal editorial boards. He served on the ERC Advanced Grants Evaluation Panel for Computer Science and Informatics and was named its Chair in 2014. He was awarded the SIAM Activity Group on Linear Algebra Prize (1994), the Annual Prize of the Academy of Sciences of the Czech Republic (2007), and the Bernard Bolzano Medal of the Academy of Sciences of the Czech Republic for Merits in Mathematical Sciences (2013). In 2014 he was selected as a SIAM Fellow for advances in numerical linear algebra, especially iterative methods. He is interested in looking for interconnections between problems and disciplines, and in viewing particular questions in a wide context.