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65N55 Numerical analysis -- Partial differential equations, boundary value problems -- Multigrid methods; domain decomposition

35J25 Partial differential equations -- Elliptic equations and systems -- Boundary value problems for second-order elliptic equations

65F50 Numerical analysis -- Numerical linear algebra -- Sparse matrices

65F35 Numerical analysis -- Numerical linear algebra -- Matrix norms, conditioning, scaling En-ligne: Springerlink | MSN | zbMath

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In this book, the author describes iterative and related methods for solving large sparse systems of linear algebraic equations. Such equations arise in the numerical solutions of partial differential equations. In order that the material in the book can be covered in a one-semester course, the author has chosen to exclude any treatment of direct methods, nonlinear equations, or eigenvalue problems. The book has a chapter on principles of linear algebra; it is, however, still required that the reader have taken courses in analysis and linear algebra.

The methods described in the book are illustrated by numerical examples based mostly on a Poisson model problem. The chapter on iterative methods has a nice section on the characterization of the convergence of linear iterations by the spectral radius of the iteration matrix. Jacobi, Gauss-Seidel and SOR iteration methods are discussed extensively. Semi-iterative and conjugate gradient methods are covered very elegantly. Naturally, the chapter on multigrid methods is very well written, since the author has made major contributions to this field. The inclusion of domain decomposition is a wise choice. In particular, the Schur complement method is presented in a very interesting manner. However, the decision to describe the algorithms in Pascal is a poor one. This could have been achieved more efficiently by using an algorithmic form. It would also have been useful to have a list of other public domain software that is currently available for handling large sparse systems. Despite these minor shortcomings, this is an excellent book and is a welcome addition to the growing literature in computational methods for iterative solution of large sparse systems of linear equations arising in the numerical solution of differential equations. (MSN)

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