Methods of solving grid equations

The book presents modern fast direct and iterative methods for solving systems of linear algebraic equations with a large number of unknowns and sparse ill-conditioned matrices. Such systems arise in the grid method for solving boundary and initial-boundary value problems for partial differential equations. The main part of the book is devoted to the construction, justification and algorithmic implementation of methods, as well as the construction of various types of preconditioners, the use of which allows to increase the efficiency of iterative methods. At the end of each chapter, information is provided that complements its content and allows the reader to navigate through a structured list of cited publications. The book contains three appendices, one of which is a summary of information from linear algebra used in various chapters. The book includes a large number of algorithms implementing these methods, and a detailed subject index.

The book will be useful to specialists who use existing effective grid methods to solve differential equations, and to researchers engaged in the development of new methods. It will be useful for university students and postgraduates specializing in numerical methods and applied mathematics.

Keywords: direct methods, iterative methods, convergence, preconditioning, grid equations, systems of linear equations, sparse matrices