Parallel preconditioners for plate and shell problems

Holger Matthes

Fakultät für Mathematik Technische Universität Chemnitz–Zwickau

D–09107 Chemnitz

            e–mail: Holger.Matthes@mathematik.tu-chemnitz.de

The solution of the bending problems of thin elastic plates and shells discretized by rectangular finite elements is considered. Domain decomposition (DD) is the basic tool for the parallelization of the conjugate gradient method as well for the construction of efficient parallel preconditioners. The ASM–DD–preconditioner proposed by Haase, Langer and Meyer is used to derive a parallel preconditioner for the finite element equations resulting from discretizations by nonconforming Adini elements. One of its main components is a fast, almost direct solver for the clamped plate problem in rectangular subdomains. The so–called boundary potential method is based on the fast Fourier transform. The other main component is a multilevel preconditioner for the Schur–complement designed for the conforming Bogner– Fox–Schmidt elements. An additional smoother for the mixed derivatives allows us to make use of this DD–preconditioner also in the case of conforming discretizations. The condition number of the preconditioned system of linear equations is bounded by a constant independent of the number of unknowns N \underline{and} independent of the number of subdomains. The resulting parallel preconditioned conjugate gradient method requires O(N ln N ln ε^{−1}) arithmetical operations per processor in order to solve the f.e. equations with the relative accuracy ε. Results of numerical experiments carried out on a multiprocessor system confirm the above mentioned theoretical results. Further basic ideas for the construction of preconditioners for the bending problem of cylindrical shells are presented.