A Coarsening Operator for the Optimal Preconditioning
            of Dirichlet and Neuman Domain Decomposition Methods:
                Application to Problems with Coefficient Jumps
                       and Bad Subdomain Aspect Ratios


                                Charbel Farhat
                 Department of Aerospace Engineering Sciences
                 and Center for Space Structures and Controls
                      University of Colorado at Boulder
                       Boulder, CO 80309-0429, U. S. A.

                                 Daniel Rixen
             Laboratoire de Techniques Aeronautiques et Spatiales
                             Universite de Liege 
                 Rue Ernest Solvay, 21, B-4000 Liege, Belgium



We present an optimal preconditioning scheme for the Dual Schur Complement and
Balancing domain decomposition methods that addresses the problems of
coefficient jumps and bad subdomain aspect ratios.  This preconditioner is
derived from energy principles and embeds a coarse problem that propagates the
error globally and accelerates convergence.  The resulting domain
decomposition method is illustrated with the solution of several realistic
large-scale problems arising from the finite element discretization of
structural and solid mechanics applications, and its optimal convergence rate
is demonstrated for arbitrary mesh partitions and heterogeneous structures.