Quasi-optimal A Priori Interface Error Bounds and A Posteriori Estimates for the Interior Penalty Method

In this work, we show quasi-optimal interface error estimates for solutions obtained by the symmetric interior penalty discontinuous Galerkin method. It is proved that the numerical solution restricted to an interface converges with order |lnh|h under suitable regularity requirements, where the logarithmic factor is only present in the lowest order case, i.e., k = 1. For this case, we also derive and analyze two a posteriori error estimators which demonstrate that the jump terms of the discrete fluxes are not essential to obtain local efficiency and reliability. We support our analysis by numerical results and demonstrate that the interface approximation can be locally postprocessed to obtain discrete solutions of order h in the energy norm.