Block Jacobi for discontinuous Galerkin discretizations: no ordinary Schwarz methods

For classical discretizations of elliptic partial differential equations, like conforming finite elements or finite differences, block Jacobi methods are equivalent to classical Schwarz methods with minimal overlap, see for example [4]. This is different when the linear system (1) is obtained using DG methods. Our paper is organized as follows: in section 2 we describe several DG methods for linear elliptic problems. We follow our discussion by introducing some “hybridizable” DG methods. In section 3 we show that block Jacobi iterations for the DG methods are corresponding to non-overlapping Schwarz methods with particular transmission conditions involving the penalty parameter of the DG method used. We then show numerical experiments in section 4, and present our conclusions in section 5.