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Discontinuous Galerkin methods for the biharmonic problem on polygonal and polyhedral meshes

Dong, Zhaonan 2019. Discontinuous Galerkin methods for the biharmonic problem on polygonal and polyhedral meshes. International Journal of Numerical Analysis and Modeling 16 (5) , pp. 825-846.

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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 special 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 Pp, 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.

Item Type: Article
Date Type: Published Online
Status: Published
Schools: Mathematics
ISSN: 1705-5105
Date of First Compliant Deposit: 22 January 2020
Last Modified: 13 Mar 2021 02:26

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