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

Zhaonan Dong
2018 arXiv   pre-print
We introduce an hp-version symmetric interior penalty discontinuous Galerkin finite element method (DGFEM) for the numerical approximation of the biharmonic equation on general computational meshes consisting of polygonal/polyhedral (polytopic) elements. In particular, the stability and hp-version a-priori error bound are derived based on the specific choice of the interior penalty parameters which allows for edges/faces degeneration. Furthermore, by deriving a new inverse inequality for a
more » ... al class of polynomial functions (harmonic polynomials), the proposed DGFEM is proven to be stable to incorporate very general polygonal/polyhedral elements with an arbitrary number of faces for polynomial basis with degree p=2,3. The key feature of the proposed method is that it employs elemental polynomial bases of total degree P_p, defined in the physical coordinate system, without requiring the mapping from a given reference or canonical frame. A series of numerical experiments are presented to demonstrate the performance of the proposed DGFEM on general polygonal/polyhedral meshes.
arXiv:1807.07817v2 fatcat:ug2uje6m3jbnfagomcugqqdjry