\(hp\)-version discontinous Gelerkin methods for advection-diffusion-reaction problems on polytropic meshes (Q2814654)

The authors analyze the \(hp\)-version of discontinous Gelerkin finite element methods for partial differential equations with nonnegative characteristic form on general meshes, consisting of polytopic elements. The general classes of polytopic elements are admitted, including elements with degenerating \((d - k)\)-dimensional facets, \(k = 1, \dots , d -1\). The underlying analysis exploits the novel \(hp\)-version approximation and inverse inequalities, together with the inf-sup condition. Numerical experiments are presented which not only confirm the theoretical results derived in this paper, but also demonstrate the efficiency of employing local polynomial spaces of total degree \(p\), defined in the physical coordinate system, compared with tensor-product polynomial bases, mapped from a given reference or canonical frame, under \(p\)-refinement.