Robust interior penalty discontinuous Galerkin methods

Classical interior penalty discontinuous Galerkin (IPDG) methods for diffusion problems require a number of assumptions on the local variation of mesh-size, polynomial degree, and of the diffusion coefficient to determine the values of the, so-called, discontinuity-penalization parameter and/or to perform error analysis. Variants of IPDG methods involving weighted averages of the gradient of the approximate solution have been proposed in the context of high-contrast diffusion coefficients to mitigate the dependence of the contrast in the stability and in the error analysis. Here, we present a new IPDG method, involving carefully constructed weighted averages of the gradient of the approximate solution, which is shown to be robust even for the most extreme simultaneous local mesh, polynomial degree and diffusion coefficient variation scenarios, without resulting in unreasonably large penalization. The new method, henceforth termed as \emph{robust IPDG} (RIPDG), offers typically significantly better conditioning than the standard IPDG method when applied to scenarios with strong mesh/polynomial degree/diffusion local variation. On the other hand, when using uniform meshes, constant polynomial degree and for problems with constant diffusion coefficients, the RIPDG method is identical to the classical IPDG. Numerical experiments indicate the favourable performance of the new RIPDG method over the classical version in terms of conditioning and error.