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

Josef Malek is a Professor at the Faculty of Mathematics and Physics, Charles University in Prague, Czech Republic. He is a director of the Necas 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.

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