Robust Multigrid Methods for Parameter Dependent Problems

Joachim Schöberl
SFB 1306, University Linz, Austria


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

( A + epsilon-1 B ) u = f,
with an elliptic operator A, an operator B with non-trivial kernel, and a small, positive parameter epsilon. We are interested in the construction and analysis of robust multigrid methods for the solution of the arising symmetric and positive definite finite element problem.

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 L2-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.