Symmetric Interior Penalty Discontinuous Galerkin Methods for Elliptic Problems in Polygons

Abstract

We analyze symmetric interior penalty discontinuous Galerkin (DG) finite element methods for linear, second-order elliptic boundary-value problems in polygons $\Omega$ with straight edges where solutions exhibit singular behavior near corners, and at boundary points where boundary conditions change. To resolve corner singularities, we admit both graded meshes and bisection refinement meshes. We prove that judiciously chosen refinement parameters in these mesh families imply optimal asymptotic rates of convergence with respect to the total number of degrees of freedom $N$, both for the DG energy norm error and the $L^2$-norm error. The sharpness of our asymptotic convergence rate estimates is confirmed in a series of numerical experiments.