Jung, Michael; Langer, Ulrich Applications of multilevel methods to practical problems. (English) Zbl 0743.65031 Surv. Math. Ind. 1, No. 3, 217-257 (1991). 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. Reviewer: H.R.Schwarz (Zürich) Cited in 9 Documents 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) Keywords:magnetic fields; thermoelasticity; mesh generation; finite elements; multilevel methods; multigrid-Newton methods; domain decomposition; parallel computation; preconditioned conjugate gradients; hierarchical bases algorithms; convergence; complexity; numerical examples PDFBibTeX XMLCite \textit{M. Jung} and \textit{U. Langer}, Surv. Math. Ind. 1, No. 3, 217--257 (1991; Zbl 0743.65031)