Shapira delivers a systematic and unified presentation of the multigrid method that is used for the efficient solution of partial differential equations. Such methodology is important to the computational science and engineering research community, and potential readers can gain a thorough insight into this topic.

The book is divided into seven parts, comprising 19 chapters. Part 1, “Concepts and Preliminaries,” has two chapters. Chapter 1 introduces the concepts of multiscale and multilevel, from basic principles to data structures, and parallel algorithms. Chapter 2 presents basic definitions and standard theoretical results that are necessary for the Fourier transform multilevel hierarchy.

Part 2, “Partial Differential Equations and Their Discretization,” also has two chapters. Chapter 3 describes the finite difference and finite volume discretization method for various types of partial differential equations. Additionally, it discusses accuracy and adequacy issues. Chapter 4 presents the finite element discretization method, including local and adaptive mesh refinement.

In Part 3, “The Numerical Solution of Large Sparse Linear Systems,” chapter 5 overviews the iterative methods for the solution of sparse linear systems, such as point and block Jacobi, Gauss-Seidel (GS), conjugate gradient methods, and incomplete LU factorizations. In chapter 6, the author describes various kinds of multigrid methods for solving sparse linear systems, highlights the connection between multigrid and domain decomposition, and addresses versions of black-box multigrid and algebraic multigrid, in terms of domain decomposition.

Part 4, “Multigrid for Structured Grids,” has five chapters. In chapter 7, the author describes the automatic multigrid iterative method that uses five-point stencils on the coarse grid, in conjunction with red-black point GS relaxation in the entire V-cycle. Chapter 8 examines the applicability of the multigrid method to image processing, in order to remove noise from grayscale and color digital images. Chapter 9 presents the black-box multigrid method for structured linear systems using nine-point stencils, which is suitable for diffusion problems with variable and discontinuous coefficients. In chapter 10, the author applies the black-box multigrid and automatic multigrid methods to the indefinite Helmholtz equation. Chapter 11 describes the matrix-based multigrid method using semi-coarsening as a combination of domain decomposition, line incomplete LU (ILU) factorization, and variational multigrid.

Part 5, “Multigrid for Semi-Structured Grids,” has two chapters. Chapter 12 presents the multigrid method using a coarse grid constructed by local refinement, to solve the linear system on the finest grid. In chapter 13, the author discusses the multigrid method based on the semi-structured grid--using finite element or volumes--that is, a combined grid consisting of the entire hierarchy of uniform grids, supplying correction terms in the V-cycle.

Part 6, “Multigrid for Unstructured Grids,” has four chapters. Chapter 14 introduces a domain decomposition two-grid iterative method for general unstructured grids that cannot be obtained from local refinement. Chapter 15 presents an algebraic multilevel method defined in terms of the coefficient matrix only. In chapter 16, the author examines the applicability of the algebraic multilevel method to diffusion problems with oblique anisotropy and highly nonsymmetric convection diffusion equations. Chapter 17 presents the semialgebraic multilevel method for systems of partial differential equations.

Part 7 contains the appendices, in two chapters. In chapter 18, the author considers time-dependent partial differential equations. The numerical solution of such a problem is obtained by solving elliptic boundary value problems at discrete-time steps (that is, a sequence of linear systems of algebraic equations). These linear systems are solved iteratively with the multigrid method. Chapter 19 discusses nonlinear boundary value problems that are linearized by the Newton method. Hence, each linear system is solved efficiently by an inner iteration of the matrix-based multigrid method. Each chapter ends with proposed exercises. Finally, the bibliography section contains 129 references on the subject, followed by an index of terms.

Shapira introduces a new approach to multigrid methods from an algebraic point of view, specifically, a unified domain decomposition approach. The notations are consistent and the presentation is self-contained.

The book is recommended to readers involved in the field of computational science and engineering, from the postgraduate to the expert level. Additionally, the book is suitable for courses in numerical analysis, numerical linear algebra, scientific computing, and numerical solution of partial differential equations.