AN EXTENSION OF THE DISCONTINUOUS GALERKIN METHOD FOR THE SINGULAR POISSON EQUATION

We present a numerical method for solving a singular Poisson equation which solution contains jumps and kinks due to a singular right-hand side. Equations of this type may arise, e. g., within the pressure computation of incompressible multiphase flows. The method is an extension to the well-known discontinuous Galerkin (DG) method, being able to represent the jumps and kinks with subcell accuracy. In the proposed method, an ansatz function which already fulfills the jump condition is subtracted from the original problem, thereby reducing it to a standard Poisson equation without a jump. Invoking a technique that we refer to as "patching," the construction of the ansatz function can be limited to a very narrow domain around the jump position, thus making the construction numerically cheap and easy. Under optimal conditions, the method shows a convergence order of p + 1 for DG polynomial degree p. Still, in the worst case, a convergence order of approximately 2.4 is preserved for DG polynomial degree of 2.