This is an excellent book on multigrid methods, written by three experts, with contributions from three others, including the father of multigrid, Achi Brandt. The book can be used by graduate students with knowledge of differential equations and the fundamentals of numerical analysis. It is also useful for scientists and engineers interested in the efficient numerical solution of differential equations modeling various realistic problems.

The first chapter gives the basics of partial differential equations, discretizations, the two ingredients of multigrid, and the iterative solution of the resulting algebraic systems. The next four chapters take the reader through the fundamentals of multigrid methods, including an analysis of nonlinear problems and higher order discretizations. Throughout the book, the reader is referred to other literature. An extensive list of references (453 in total) is given. Chapter 6 details parallelism of multigrid solvers and modifications to multigrid for obtaining more efficiency from parallelism.

More advanced topics, such as convection-diffusion equations, problems with mixed derivatives and discontinuous coefficients are described in chapter 7. Multigrid for systems of differential equations such as the incompressible Navier-Stokes and compressible Euler equations are discussed in chapter 8. More on multigrid for fluid flow problems is given in chapter 10. Adaptive refinements of multigrid for problems on L-shaped domains and nonlinear problems with a shock are discussed in chapter 9.

The book closes with three appendices. The first, written by K. Stüben, is on algebraic multigrid, that is, the given problem is not defined on a grid, but rather the method operates directly on sparse linear algebraic equations. Applications to computational fluid dynamics are given. The second appendix, written by P. Oswald, gives the theoretical underpinnings of iterative subspace correction methods for the solution of symmetric positive definite variational problems. The convergence properties of additive and multiplicative Schwarz methods are given with applications to multigrid and domain decomposition.

The last appendix, written by the father of multigrid, is probably the most important for research into achieving efficient multigrid solvers for fluid dynamic problems. It gives a table listing open questions, possible solutions and the status at the time the appendix was written. We quote “The table deals only with steady state flows and their direct multigrid solvers...Time accurate solvers for genuine time-dependent flow problems are in principle simpler to develop than their steady state counterpart.”