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Applications of multilevel methods to practical problems. (English) Zbl 0743.65031

The paper gives a nice review of some multilevel methods that are well suited for the solution of real-life linear and nonlinear boundary value problems. The paper starts with describing a first version of a hierarchical mesh generator and an improved version for interface problems. Then a survey of multilevel algorithms is given including multigrid methods, multigrid preconditioned conjugate gradients, hierarchical bases algorithms and domain decomposition methods.
The corresponding convergence and complexity results are presented and especially the advantages and disadvantages of the various methods in connection with real-life problems with complicated geometries, interfaces and singularities of the solutions are discussed.
Finally, the solution of nonlinear problems by multigrid Newton techniques is described and numerical examples from magnetic and thermomechanical field problems are given.

MSC:

65F10 Iterative numerical methods for linear systems
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65Y05 Parallel numerical computation
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
74A15 Thermodynamics in solid mechanics
78A25 Electromagnetic theory (general)
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