(A) dual iterative substructuring method with a penalty term

Abstract

In the methods of non-overlapping domain decomposition for second-order elliptic problems, it is well-known that the weak solution is equivalent to the minimizer of the energy functional defined by a sum of the local energy on partitioned subdomain if the trace continuity constraint across the interface is satisfied. Based on the constrained minimization, many studies for treatment of the continuity constraint have been done in view of various perspectives: the Lagrangian method, the method of penalty function, and the augmented Lagrangian method, etc. The Primal-Dual FETI (FETI-DP) method is one of the most advanced dual substructuring methods, which partially follows the principle of the Lagrangian method. The FETI-DP enforces the continuity on the interface in two different points of view: the continuity of the primal solution at corners caused by making all of the associated subdomains with a corner share the same degree of freedom at the corner and the continuity on the rest of the interface imposed by using Lagrange multipliers following the standard Lagrangian approach for constrained minimizations. In this dissertation, an iterative substructuring method with Lagrange multipliers is considered for second order elliptic problems in both two dimensions and three dimensions. We first propose a dual substructuring algorithm with a penalty term in two dimensions. The proposed method treats the continuity constraint across the interface in view of the augmented Lagrangian method, that is, imposes the continuity constraint by augmenting a penalty term as well as the pointwise matching condition on the interface in the same way as in the FETI-DP. To the Lagrangian functional, we add a penalty term which measures the jump across the interface and includes a positive penalization parameter $\eta$. As in most dual substructuring methods, the original problem is reduced to a dual system for Lagrange multipliers by eliminating all of primal degrees of freedom in...