This classic text presents the best practices of developing multigrid solvers for large-scale computational problems in science and engineering. By representing a problem at multiple scales and employing suitable interscale interactions, multigrid avoids slowdown due to stiffness and reduces the computational cost of classical algorithms by orders of magnitude. Starting from simple examples, this book guides the reader through practical stages for developing reliable multigrid solvers, methodically supported by accurate performance predictors. The revised edition presents discretization and fast solution of linear and nonlinear partial differential systems; treatment of boundary conditions, global constraints and singularities; grid adaptation, high-order approximations and system design optimization; applications to fluid dynamics, from simple models to advanced systems; new quantitative performance predictors, a MATLAB ® sample code and more. Readers will also gain access to the Multigrid Guide 2.0 website, where updates and new developments will be continually posted, including a chapter on Algebraic Multigrid.