The Two-Level Feti Method for Static and Time Dependent Fourth-Order PDEs:
Application to Plate and Shell Problems 

Charbel Farhat, Po-Shu Chen                       Jan Mandel
Department of Aerospace Engineering Sciences      Center for Computational Mathematics
and Center for Aerospace Structures               University of Colorado at Denver
University of Colorado at Boulder                 Denver, CO 80217-3364, U. S. A.
Boulder, CO 80309-0429, U. S. A.                  and Department of Aerospace Engineering Sciences
                                                  University of Colorado at Boulder
                                                  Boulder, CO 80309-0429, U. S. A.
We present a Lagrange multiplier based substructuring method for solving iteratively large-scale systems of equations arising from the finite element discretization of static and dynamic plate and shell pro blems. The proposed method is essentially an extension of the FETI domain decomposition algorithm to fourth-order problems. The main idea is to enforce exactly the continuity of the transverse displacement field at the substructure corners throughout the preconditioned conjugate projected gradient iterations. This results in a two-level FETI substructuring method where the condition number of the preconditioned interface problem does not grow with the number of substructures, and grows at most polylogarithmically with the number of elements per substructure. These theoretically proven optimal convergence properties of the new FETI method are numerically demonstrated for several finite element plate and shell static and transient problems. The two-level iterative solver presented in this paper is applicable to a large family of biharmonic time-independent as well as time-dependent systems.

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