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AuthorShapira, Yair. author
TitleMatrix-Based Multigrid [electronic resource] : Theory and Applications / by Yair Shapira
ImprintBoston, MA : Springer US : Imprint: Springer, 2003
Connect tohttp://dx.doi.org/10.1007/978-1-4757-3726-4
Descript XVI, 221 p. 6 illus. online resource

SUMMARY

Many important problems in applied science and engineering, such as the Navierยญ Stokes equations in fluid dynamics, the primitive equations in global climate modยญ eling, the strain-stress equations in mechanics, the neutron diffusion equations in nuclear engineering, and MRIICT medical simulations, involve complicated sysยญ tems of nonlinear partial differential equations. When discretized, such problems produce extremely large, nonlinear systems of equations, whose numerical solution is prohibitively costly in terms of time and storage. High-performance (parallel) computers and efficient (parallelizable) algorithms are clearly necessary. Three classical approaches to the solution of such systems are: Newton's method, Preconditioned Conjugate Gradients (and related Krylov-space acceleration techยญ niques), and multigrid methods. The first two approaches require the solution of large sparse linear systems at every iteration, which are themselves often solved by multigrid methods. Developing robust and efficient multigrid algorithms is thus of great importance. The original multigrid algorithm was developed for the Poisson equation in a square, discretized by finite differences on a uniform grid. For this model problem, multigrid exhibits extremely rapid convergence, and actually solves the problem in the minimal possible time. The original algorithm uses rediscretization of the partial differential equation (POE) on each grid in the hierarchy of coarse grids that are used. However, this approach would not work for more complicated problems, such as problems on complicated domains and nonuniform grids, problems with variable coefficients, and non symmetric and indefinite equations. In these cases, matrix-based multi grid methods are in order


CONTENT

1. The Multilevel-Multiscale Approach -- I The Problem and Solution Methods -- 2. PDEs and Discretization Methods -- 3. Iterative Linear-System Solvers -- 4. Multigrid Algorithms -- II Multigrid for Structured Grids -- 5. The Automug Method -- 6. Applications in Image Processing -- 7. The Black-Box Multigrid Method -- 8. The Indefinite Helmholtz Equation -- 9. Matrix-Based Semicoarsening -- III Multigrid for Semi-Structured Grids -- 10. Multigrid for Locally Refined Meshes -- IV Multigrid for Unstructured Grids -- 11. Domain Decomposition -- 12. Algebraic Multilevel Method -- 13. Conclusions -- Appendices -- A C++ Framework for Unstructured Grids -- References


Mathematics Computer science -- Mathematics Numerical analysis Applied mathematics Engineering mathematics Computer mathematics Mathematics Computational Mathematics and Numerical Analysis Numeric Computing Mathematics of Computing Applications of Mathematics



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