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The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids. (English) Zbl 0857.65129
For the numerical solution of elliptic boundary value problems discretized by finite element schemes on unstructured grids the author develops an abstract framework of the auxiliary space method. This is an optimal preconditioning technique that results from the proper combination of an appropriate smoother and a structured auxiliary grid and can be interpreted as a two-level nonnested multigrid preconditioner. Examples include the use of a
– nested coarse grid space as an auxiliary space,
– lower-order element as an auxiliary space for a higher-order element,
– conforming element as an auxiliary space for a nonconforming element.

MSC:
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65F35 Numerical computation of matrix norms, conditioning, scaling
65F10 Iterative numerical methods for linear systems
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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