Robust Multigrid Methods for Parameter Dependent Problems

Several models from computational mechanics lead to

parameter dependent problems of the form

with an elliptic operator A, an operator B with non-trivial

kernel, and a small, positive parameter

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interested in the construction and analysis of robust multigrid

methods for the solution of the arising symmetric and positive

definite linear systems.

Specific examples considered in this talk are nearly incompressible

materials and the Reissner Mindlin plate model. We shortly present

robust non-conforming discretization schemes equivalent to

corresponding mixed finite element methods.

We will explain the necessary multigrid components, namely a block

smoother covering basis function of the kernel of B, and

prolongation operators mapping coarse-grid kernel functions to

fine-grid kernel functions.

We present the analysis for the two-level algorithm based on the

Additive Schwarz Technique. Optimal and robust convergence rates of

the multigrid W-cycle and variable V-cycle algorithms are proved by the

smoothing property and the approximation property. We have to use

norms depending in a proper way on the small parameter

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Numerical experiments indicate also optimal convergence rates for

the V-cycle algorithm for both problems.