  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.

Numerical Analysis;Numerical Analysis