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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 Web site, where updates and new developments will be continually posted, including a chapter on Algebraic Multigrid. Audience: This book will be useful to practitioners and researchers, as well as students and instructors, in many areas of computational science and engineering, applied mathematics, and numerical analysis. Contents: Chapter 0: Introduction; Chapter 1: Elementary Acquaintance with Multigrid; Part I: Stages in Developing Fast Solvers; Chapter 2: Stable Discretization; Chapter 3: Interior Relaxation and Smoothing Factors; Chapter 4: Interior Two-Level Cycles; Chapter 5: Boundary Conditions and Two-Level Cycling; Chapter 6: Many-Level Cycles; Chapter 7: Full Multi-Grid (FMG) Algorithms; Part II: Advanced Techniques and Insights; Chapter 8: Full Approximation Scheme (FAS) and Applications; Chapter 9: Local Refinements and Grid Adaptation; Chapter 10: Higher-Order Techniques; Chapter 11: Coarsening Guided By Discretization; Chapter 12: True Role of Relaxation; Chapter 13: Dealgebraization of Multigrid; Chapter 14: Practical Role of Rigorous Analysis and Quantitative Predictions; Chapter 15: Chains of Problems. Frozen T; Chapter 16: Time Dependent Problems; Part II: Applications to Fluid Dynamics; Chapter 17: Cauchy-Riemann Equations; Chapter 18: Steady-State Stokes Equations; Chapter 19: Steady-State Incompressible Navier-Stokes Equations; Chapter 20: Compressible Navier-Stokes and Euler Equations; Chapter 21: Remarks on Solvers for Transonic Potential Equations; Appendix: Test Cycle: MATLAB Code