A Divergence-Free $P_k$ CDG Finite Element for the Stokes Equations on Triangular and Tetrahedral Meshes
Year: 2025
Author: Xiu Ye, Shangyou Zhang
Numerical Mathematics: Theory, Methods and Applications, Vol. 18 (2025), Iss. 1 : pp. 157–174
Abstract
In the conforming discontinuous Galerkin method, the standard bilinear form for the conforming finite elements is applied to discontinuous finite elements without adding any inter-element nor penalty form. The $P_k$ $(k ≥ 1)$ discontinuous finite elements and the $P_{k−1}$ weak Galerkin finite elements are adopted to approximate the velocity and the pressure respectively, when solving the Stokes equations on triangular or tetrahedral meshes. The discontinuous finite element solutions are divergence-free and surprisingly H-div functions on the whole domain. The optimal order convergence is achieved for both variables and for all $k ≥ 1.$ The theory is verified by numerical examples.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/nmtma.OA-2024-0063
Numerical Mathematics: Theory, Methods and Applications, Vol. 18 (2025), Iss. 1 : pp. 157–174
Published online: 2025-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 18
Keywords: Finite element conforming discontinuous Galerkin method Stokes equations stabilizer free divergence free.