Corners and stable optimized domain decomposition methods for the Helmholtz problem

We construct a new Absorbing Boundary Condition (ABC) adapted to solving the Helmholtz equation in polygonal domains in dimension two. Quasi-continuity relations are obtained at the corners of the polygonal boundary. This ABC is then used in the context of domain decomposition where various stable algorithms are constructed and analysed. Next, the operator of this ABC is adapted to obtain a transmission operator for the Domain Decomposition Method (DDM) that is well suited for broken line interfaces. For each algorithm, we show the decrease of an adapted quadratic pseudo-energy written on the skeleton of the mesh decomposition, which establishes the stability of these methods. Implementation within a finite element solver (GMSH/GetDP) and numerical tests illustrate the theory.