Dealing with cross-points in a non-overlapping domain decomposition solution of the Helmholtz equation

When coupling a finite element and a domain decomposition method, a cross-point corresponds to a degree of freedom shared by more than two domains. The problem of dealing with such cross-points is addressed for the case of an usual nodal finite element solution of the Helmholtz equation. An important feature of the approach relies upon its interpretation as an iterative method for solving the discrete problem in the whole domain. The convergence of the iterative procedure is established in the general case and proved to be scalable, that is, to converge at a rate independent of the mesh when the domain decomposition method involves no cross-points.