THE DUAL SCHUR COMPLEMENT METHOD WITH WELL-POSED LOCAL NEUMANN PROBLEMS - REGULARIZATION WITH A PERTURBED LAGRANGIAN FORMULATION

The dual Schur complement (DSC) domain decomposition (DD) method intro duced by Farhat and Roux is an efficient and practical algorithm for t he parallel solution of self-adjoint elliptic partial differential equ ations. A given spatial domain is partitioned into disconnected subdom ains where an incomplete solution for the primary field is first evalu ated using a direct method. Next, intersubdomain field continuity is e nforced via a combination of discrete, polynomial, and/or piece-wise p olynomial Lagrange multipliers, applied at the subdomain interfaces. T his leads to a smaller size symmetric dual problem where the unknowns are the ''gluing'' Lagrange multipliers, and which is best solved with a preconditioned conjugate gradient (PCG) algorithm. However, for tim e-independent elasticity problems, every floating subdomain is associa ted with a singular stiffness matrix, so that the dual interface opera tor is in general indefinite. Previously, we have dealt with this issu e by filtering out at each iteration of the PCG algorithm the contribu tions of the local null spaces. We have shown that for a small number of subdomains, say less than 32, this approach is computationally feas ible. Unfortunately, the filtering phase couples the subdomain computa tions, increases the numerical complexity of the overall solution algo rithm, and limits its parallel implementation scalability, and therefo re is inappropriate for a large number of subdomains. In this paper, w e regularize the DSC method with a perturbed Lagrangian formulation wh ich restores the positiveness of the dual interface operator, reduces the computational complexity of the overall methodology, and improves its parallel implementation scalability. This regularization procedure corresponds to a novel splitting method of the interface operator whi ch entails well-posed local discrete Neumann problems, even in the pre sence of floating subdomains. Therefore, it can also be interesting fo r other DD algorithms such as those considered by Bjorstad and Widlund , Marini and Quarteroni, De Roeck and Le Tallec, and recently by Mande l.