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We introduce an $hp$-version symmetric interior penalty discontinuousGalerkin finite element method (DGFEM) for the numerical approximation of thebiharmonic equation on general computational meshes consisting ofpolygonal/polyhedral (polytopic) elements. In particular, the stability and$hp$-version a-priori error bound are derived based on the specific choice ofthe interior penalty parameters which allows for edges/faces degeneration.Furthermore, by deriving a new inverse inequality for a special class {of}polynomial functions (harmonic polynomials), the proposed DGFEM is proven to bestable to incorporate very general polygonal/polyhedral elements with an\emph{arbitrary} number of faces for polynomial basis with degree $p=2,3$. Thekey feature of the proposed method is that it employs elemental polynomialbases of total degree $\mathcal{P}_p$, defined in the physical coordinatesystem, without requiring the mapping from a given reference or canonicalframe. A series of numerical experiments are presented to demonstrate theperformance of the proposed DGFEM on general polygonal/polyhedral meshes.

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