Exponential convergence of the \(hp\)-DGFEM for diffusion problems (Q1827216)

Two different formulations of the \(hp\)-discontinuous Galerkin finite element method (DGFEM) are considered for the two-dimensional stationary diffusion problem. As in the usual FEM, mesh refinement strategy is important when corner singularity caused by polygonal shape of domains exists. The authors prove exponential convergence of the \(hp\)-version of DGFEM on geometrically refined meshes in polygons. Several variants of interior penalization are covered. Numerical experiments indicate the sharpness of the theoretical results as well as the weak dependence of the DGFEM approximation on the particular choice of interior penalization and the penalty parameter. In certain cases, stabilization techniques are effective.