Relaxing the Roles of Corners in BDDC by Perturbed Formulation

摘要

The Balancing Domain Decomposition by Constraints (BDDC) method was first introduced by Dohrmann (2003). Compared to its parent, the BDD method by Mandel (1993), one of the advances in BDDC method is the use of constraints to enforce equality of averages across faces, edges, or at individual dofs on substructure boundaries called corners. These constraints serve two purposes. First, they ensure that the coefficient matrix of the coarse problem is always invertible. Second, they induce a natural coarse space leading to fast convergence. While corner constraints do not have significant contribution in serving the second purpose, they are mainly responsible for the first one. In addition, in order to use positive definite sparse direct solvers, which are faster and more robust than their indefinite counterparts, the corners should be chosen so that the local matrix sub-assembled for all dofs in each substructure except corners is positive definite. Here we do not consider a change of basis, cf. Li and Widlund (2006), as it destroys good sparsity pattern of local matrices and is more complicated to implement.