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Reviewer: Beny Neta

As the title suggests, this monograph gives the fundamentals of the multigrid method of solving partial differential equations. In ten chapters, the authors take the reader from the basic concepts of multigrid methods, based on finite difference solution of linear problems, to a discussion of nonlinear problems and even finite element–based methods. The first chapter presents model problems and their finite difference approximation. The second chapter gives basic iterative methods for the solution of the algebraic equations resulting from the discretization. Multigrid methods are introduced in chapter 3. In the next chapter, the implementation of multigrid is discussed. The monograph does not include code, but refers the reader to a Web page that offers at least 22 free software packages. Chapter 5 presents a bit of the theory. Mathematical proofs are not given. Nonlinear problems and iterative processes for the solution of the nonlinear systems resulting from the discretization are presented in chapter 6. Chapter 7 discusses three different applications, including the case of variable coefficients. The next two chapters present the algebraic multigrid method (for purely algebraic problems such as networks) and the multilevel adaptive methods. The monograph closes with an example of using multigrid methods when the algebraic equations result from finite element discretization. This book is a valuable reference for those interested in multigrid methods. Together with the free software on the Web, it clearly shows the power of multigrid methods in a variety of applications. Several interesting applications—such as time-dependent partial differential equations and integral equations—are not discussed here, but could be given in a follow-up book.