Robust Multigrid Methods for Parameter Dependent Problems

Several models from computational mechanics lead to parameter dependent problems of the form $$(A+\frac{1}{ϵ}B)u=f$$ with an elliptic operator $A$, an operator $B$ with non-trivial kernel, and a small, positive parameter $ϵ$;. We are 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 ε.
Numerical experiments indicate also optimal convergence rates for the V-cycle algorithm for both problems.