Volume 16, Issue 5
Discontinuous Galerkin Methods for the Biharmonic Problem on Polygonal and Polyhedral Meshes
DOI:

Int. J. Numer. Anal. Mod., 16 (2019), pp. 825-846.

Published online: 2019-08

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• Abstract

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 $\mathcal{P}$$p, defined in the physical coordinate system, without requiring the mapping from a given reference or canonical frame. A series of numerical experiments are presented to demonstrate the performance of the proposed DGFEM on general polygonal/polyhedral meshes. • Keywords Discontinuous Galerkin, polygonal/polyhedral elements, inverse estimates, biharmonic problems. • AMS Subject Headings 65N30, 65N50, 65N55 • Copyright COPYRIGHT: © Global Science Press • Email address zd14@le.ac.uk (Zhaonan Dong) • BibTex • RIS • TXT @Article{IJNAM-16-825, author = {Dong , Zhaonan }, title = {Discontinuous Galerkin Methods for the Biharmonic Problem on Polygonal and Polyhedral Meshes}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2019}, volume = {16}, number = {5}, pages = {825--846}, abstract = { We introduce an hp-version symmetric interior penalty discontinuous Galerkin finite element method (DGFEM) for the numerical approximation of the biharmonic equation on general computational meshes consisting of polygonal/polyhedral (polytopic) elements. In particular, the stability and hp-version a-priori error bound are derived based on the specific choice of the interior penalty parameters which allows for edges/faces degeneration. Furthermore, by deriving a new inverse inequality for a special class of polynomial functions (harmonic polynomials), the proposed DGFEM is proven to be stable to incorporate very general polygonal/polyhedral elements with an arbitrary number of faces for polynomial basis with degree p = 2, 3. The key feature of the proposed method is that it employs elemental polynomial bases of total degree \mathcal{P}$$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.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/13256.html} }
TY - JOUR T1 - Discontinuous Galerkin Methods for the Biharmonic Problem on Polygonal and Polyhedral Meshes AU - Dong , Zhaonan JO - International Journal of Numerical Analysis and Modeling VL - 5 SP - 825 EP - 846 PY - 2019 DA - 2019/08 SN - 16 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/13256.html KW - Discontinuous Galerkin, polygonal/polyhedral elements, inverse estimates, biharmonic problems. AB -

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 $\mathcal{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.

Zhaonan Dong. (2019). Discontinuous Galerkin Methods for the Biharmonic Problem on Polygonal and Polyhedral Meshes. International Journal of Numerical Analysis and Modeling. 16 (5). 825-846. doi:
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