SFB 1306, University Linz, Austria

Several problems in computational mechanics lead to parameter dependent problems of the form

Specific examples considered in this talk are nearly incompressible materials and Reissner Mindlin plate models. 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. For a
large class of problems including Reissner Mindlin models we can prove optimal
iteration numbers for a two-level method by an abstract Lemma.

For the example of nearly incompressible materials we present new results for
optimal W-cycle convergence rate estimates. Key components are a
L_{2}-like norm depending on the parameter *epsilon*, for which
the approximation property and smoothing property hold true uniformly in
*epsilon*. The approximation property involves coarse grid approximation
as well as approximation in the grid transfer steps. For the proof of the
smoothing property we have to use an interpolation norm between the problem
energy norm and the energy norm of the block Jacobi preconditioner. In
addition to the obtained results, we will also discuss the missing parts for
V-cycle estimates. Numerical experiments indicate optimal rates for the
V-cycle, for the problem of nearly incompressible materials and for Reissner
Mindlin plate models as well.