\documentstyle{article} \begin{document} 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. \end{document}