Year: 2023
Author: Xiu Ye, Shangyou Zhang
CSIAM Transactions on Applied Mathematics, Vol. 4 (2023), Iss. 2 : pp. 256–274
Abstract
It is known that discontinuous finite element methods use more unknown variables but have the same convergence rate comparing to their continuous counterpart. In this paper, a novel conforming discontinuous Galerkin (CDG) finite element method is introduced for Poisson equation using discontinuous $P_k$ elements on triangular and tetrahedral meshes. Our new CDG method maximizes the potential of discontinuous $P_k$ element in order to improve the convergence rate. Superconvergence of order two for the CDG finite element solution is proved in an energy norm and in the $L^2$ norm. A local post-process is defined which lifts a $P_k$ CDG solution to a discontinuous $P_{k+2}$ solution. It is proved that the lifted $P_{k+2}$ solution converges at the optimal order. The numerical tests confirm the theoretic findings. Numerical comparison is provided in 2D and 3D, showing the $P_k$ CDG finite element is as good as the $P_{k+2}$ continuous Galerkin finite element.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/csiam-am.SO-2021-0051
CSIAM Transactions on Applied Mathematics, Vol. 4 (2023), Iss. 2 : pp. 256–274
Published online: 2023-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 19
Keywords: Finite element conforming discontinuous Galerkin method stabilizer free triangular grid tetrahedral grid.
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