Non-overlapping Domain Decomposition Method and Nodal Finite Element Method

The non-overlapping domain decomposition method is an efficient approach for solving time harmonic scattering wave problems. It is used here to reduce large size systems solution to that of several systems of small size and to construct efficient procedures to couple finite element and boundary element methods. The lack of a satisfactory treatment of the so-called cross-points, nodes being shared by more than two domains, prevents one from taking advantage of the simplicity of the standard finite element method and the effectiveness of a domain decomposition procedure at the same time. A new approach overcoming this difficulty is introduced. It mainly consists in keeping a strong coupling at cross-points by enforcing a strong continuity requirement at these points for both trial and the test functions. 1. Introduction. Several methods have been devised in the last couple of years to solve the large size linear systems arising from the discretization of time harmonic scattering problems (see for example [26, 14, 19, 12, 22]). This is primarily because of the oscillatory character of the solution which accordingly requires resorting to very refined meshes. In addition, the lack of strong coercive properties of the underlaying equations, as compared for instance to those occurring in structural mechanics problems, seriously damages the efficiency of the usual solvers. This partly explains why various domain decomposition techniques have been proposed to deal with such a class of problems The aim of this paper is to contribute to this circle of techniques. We devise a new approach for the cross-points, that is, points being shared by more that two subdomains, reducing the domain decomposition procedure to a simple and efficient iterative method for solving the nodal equations. We then show that this method adapts easily to the coupling BEM-FEM algorithm. In addition, we introduce an algorithm called " evanescent modes damping algorithm " [9, 8], in order to suitably treat the evanescent part of the solution and thus improve the convergence of the domain decomposition method. The techniques presented in this paper are described in the framework of the treatment of the Helmholtz equation using the non-overlapping domain decomposition method originally introduced by P.-L. Lions [25] for solutions of the Laplace equation. It was subsequently extended to time harmonic wave propagation problems by B. Després [16, 18]. In this way, we can illustrate the several advantages it owns. It reduces the large size system solution to that of …