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PARALLEL ADDITIVE AND MULTIPLICATIVE PRECONDITIONERS
FOR PROBLEMS OF THIN SHELLS
Michael Thess
Department of Mathematics,
Technical University of Chemnitz, Germany
E-Mail: m.thess@mathematik.tu-chemnitz.de
Multilevel preconditioners are very efficient for second-order elliptic finite
element discretizations, since they have a convergence rate which is independent
of the discretization parameter, and the cost of arithmetical work per iteration
step is proportional to the number of unknowns. We distinguish between additive
multilevel preconditioners like BPX and multiplicative multilevel preconditioners
like multigrid-based preconditioners. Additive methods usually possess a lower
convergence rate but are better suited for parallelization than multiplicative
ones.
Using the results of P. Oswald for the biharmonic problem, in this talk we
describe the construction of BPX and Multilevel Diagonal Scaling (MDS-BPX)
preconditioners for the elasticity problem of thin smooth shells of arbitrary
forms where we use Koiter's shell model. The discretization is based on conforming
$C^1$ Bogner-Fox-Schmit (BFS) elements and the preconditioned conjugate gradient method serves as iterative method. On the other hand, using the estimations of the
stability of the multilevel subspace splitting, we also construct a multigrid-like
preconditioner which can be considered as multiplicative counterpart to the
MDS-BPX.
We describe the algorithms of the MDS-BPX and the multigrid preconditioner as
well as the parallelization concept which is based on a non-overlapping domain
decomposition (DD) data structure. Some implementation details of the
preconditioners on MIMD parallel computers are mentioned. The results of both
preconditioners for plates, arches, spheres, hyperbolids, and a more complicated
structure are presented by means of different numerical examples. In all cases
the iteration numbers became nearly constant for decreasing mesh sizes but depend
on the geometry parameters of the shell. For shells of simple geometry the
additive is faster than the multiplicative preconditioner. By contrast, for
shells of complicated forms or very small thicknesses the multigrid
preconditioner outperforms the MDS-BPX even in the many processor case.
Then we present an optimal DD method of the Dirichlet-Dirichlet type.
In the subdomains we use the multilevel preconditioners described above,
on the interface also an MDS-BPX like preconditioner for the Schur complement
is used. A cruical rule plays the extension operator from the FE space of the
interface into the interior of the subdomains. For second order boundary value
problems and linear basis functions S. Nepomnyaschikh has constructed such an
optimal extension operator using BPX-like techniques which we adjust to the shell
problems discretized by BFS elements. For different shells we compare the
iteration numbers of the DD preconditioner with those of the global multilevel
preconditioners which give better results. Finally, we present an outview
to the construction od DD preconditioners for problems of junctions of shells.
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