Domain Decomposition Algorithms for an Indefinite Hypersingular Integral Equation in Three Dimensions

In this paper we report on a non-overlapping and an overlapping domain decomposition method as preconditioners for the boundary element approximation of an indefinite hypersingular integral equation on a surface. The equation arises from an integral reformulation of the Neumann screen problem with the Helmholtz equation in the exterior of a screen in R. It is well-known that the linear algebraic system arising from the boundary element approximation to this integral equation is indefinite, and an iterative method like GMRES can be used to solve the system. Preconditioners by domain decomposition methods can be used to reduce the number of iterations. A non-overlapping preconditioner for the hypersingular integral equation reformulation of the 2D problem is studied in [10]. In this paper we study both non-overlapping and overlapping methods for the 3D problem. We prove that the convergence rate depends logarithmically on H/h for the non-overlapping method, and on H/δ for the overlapping method, where H and h are respectively the size of the coarse mesh and fine mesh, and δ is the overlap size. We note that domain decomposition methods with finite element approximations for the Helmholtz equation have been studied by many authors; see e.g. [2, 3, 5].