Domain decomposition method and nodal finite element for solving Helmholtz equation

The utilization of a non-overlapping domain decomposition method, in the framework of a resolution by finite elements, requires a particular treatment of the degrees of freedom shared by more than two subdomains. This is the case, for example, when solving a Laplace or Helmholtz equation by means of a conformal nodal finite element method. For convenience, such degrees of freedom will be called 'cross-points'. We describe here an approach permitting such a treatment. In contrast to a domain decomposition method in the strict sense, our approach requires a post-processing completing each iteration, which consists of solving a system whose size is the number of cross-points. We prove that the algorithm cannot break down and that it converges.