Discontinuous Galerkin Methods for the Biharmonic Problem on Polygonal and Polyhedral Meshes

Abstract

We introduce an $hp$-version symmetric interior penalty discontinuous Galerkin finite\r element method (DGFEM) for the numerical approximation of the biharmonic equation on general\r computational meshes consisting of polygonal/polyhedral (polytopic) elements. In particular, the\r stability and $hp$-version a-priori error bound are derived based on the specific choice of the interior\r penalty parameters which allows for edges/faces degeneration. Furthermore, by deriving a new\r inverse inequality for a special class of polynomial functions (harmonic polynomials), the proposed\r DGFEM is proven to be stable to incorporate very general polygonal/polyhedral elements with\r an $arbitrary$ number of faces for polynomial basis with degree $p$ = 2, 3. The key feature of the\r proposed method is that it employs elemental polynomial bases of total degree $\mathcal{P}$_$p$, defined in the\r physical coordinate system, without requiring the mapping from a given reference or canonical\r frame. A series of numerical experiments are presented to demonstrate the performance of the\r proposed DGFEM on general polygonal/polyhedral meshes.