Preconditioning the Poincaré-Steklov operator by using Green's function

This paper is concerned with the Poincaré-Steklov operator that is widely used in domain decomposition methods. It is proved that the inverse of the Poincaré-Steklov operator can be expressed explicitly by an integral operator with a kernel being the Green's function restricted to the interface. As an application, for the discrete Poincaré-Steklov operator with respect to either a line (edge) or a star-shaped web associated with a single vertex point, a preconditioner can be constructed by first imbedding the line as the diameter of a disk, or the web as a union of radii of a disk, and then using the Green's function on the disk. The proposed technique can be effectively used in conjunction with various existing domain decomposition techniques, especially with the methods based on vertex spaces (from multi-subdomain decomposition). Some numerical results are reported.