Convergence analysis of Schwarz methods without overlap for the Helmholtz equation

In this paper, the continuous and discrete optimal transmission conditions for the Schwarz algorithm without overlap for the Helmholtz equation are studied. Since such transmission conditions lead to non-local operators, they are approximated through two different approaches. The first approach, called optimized, consists of an approximation of the optimal continuous transmission conditions with partial differential operators, which are then optimized for efficiency. The second approach, called approximated, is based on pure algebraic operations performed on the optimal discrete transmission conditions. After demonstrating the optimal convergence properties of the Schwarz algorithm new numerical investigations are performed on a wide range of unstructured meshes and arbitrary mesh partitioning with cross points. Numerical results illustrate for the first time the effectiveness, robustness and comparative performance of the optimized and approximated Schwarz methods on a model problem and on industrial problems.     

Adaptive selection of face coarse degrees of freedom in the BDDC and the FETI-DP iterative substructuring methods

We propose a class of methods for the adaptive selection of the coarse space of the BDDC and FETI-DP iterative substructuring methods. The methods work by adding coarse degrees of freedom constructed from eigenvectors associated with intersections of selected pairs of adjacent substructures. It is assumed that the starting coarse degrees of freedom are already sufficient to prevent relative rigid body motions in any selected pair of adjacent substructures. A heuristic indicator of the the condition number is developed and a minimal number of coarse degrees of freedom is added to decrease the indicator under a given threshold. It is shown numerically on 2D elasticity problems that the indicator based on pairs of substructures with common edges predicts the actual condition number reasonably well, and that the method can select adaptively the hard part of the problem and concentrate computational work there to achieve good convergence of the iterations at a modest cost.