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 introduced by Farhat and Roux is an efficient and practical algorithm for the parallel solution of self-adjoint elliptic partial differential equations. A given spatial domain is partitioned into disconnected subdomains where an incomplete solution for the primary field is first evaluated using a direct method. Next, intersubdomain field continuity is enforced via a combination of discrete, polynomial, and/or piece-wise polynomial Lagrange multipliers, applied at the subdomain interfaces. This 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 time-independent elasticity problems, every floating subdomain is associated with a singular stiffness matrix, so that the dual interface operator is in general indefinite. Previously, we have dealt with this issue by filtering out at each iteration of the PCG algorithm the contributions of the local null spaces. We have shown that for a small number of subdomains, say less than 32, this approach is computationally feasible. Unfortunately, the filtering phase couples the subdomain computations, increases the numerical complexity of the overall solution algorithm, and limits its parallel implementation scalability, and therefore is inappropriate for a large number of subdomains. In this paper, we regularize the DSC method with a perturbed Lagrangian formulation which 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 which entails well-posed local discrete Neumann problems, even in the presence of floating subdomains. Therefore, it can also be interesting for other DD algorithms such as those considered by Bjorstad and Widlund, Marini and Quarteroni, De Roeck and Le Tallec, and recently by Mandel.