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Multilevel methods as iteration processes on generating systems. (Multilevelmethoden als Iterationsverfahren über Erzeugendensystemen.) (German) Zbl 0823.65026

Teubner Skripten zur Numerik. Stuttgart: B. G. Teubner. viii, 175 S. (1994).
The author introduces the idea of generating systems for the representation of functions and the discretization of elliptic boundary value problems. By using the Ritz-Galerkin ansatz for generating systems a semi-definite system of equations results. This concept yields a better understanding of modern multilevel methods.
Traditional iterative methods (Gauss-Seidel, conjugate gradients) can be interpreted as multilevel methods (multigrid, BPX) for the solution of the corresponding definite problems on the finest grid level. Furthermore this new approach can be used to construct new multilevel methods (multiple-coarse-grid, sparse grids) and adaptive algorithms.
It will be interesting to see how this new approach works for the Stokes or Navier-Stokes equations.

MSC:

65F10 Iterative numerical methods for linear systems
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
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