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.
You do not have full access to this article.
Already a Subscriber? Sign in as an individual or via your institution
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.
Author Details
-
Constructing a CDG Finite Element with Order Two Superconvergence on Rectangular Meshes
Ye, Xiu | Zhang, ShangyouCommunications on Applied Mathematics and Computation, Vol. (2023), Iss.
https://doi.org/10.1007/s42967-023-00330-5 [Citations: 0] -
Two-Order Superconvergent CDG Finite Element Method for the Heat Equation on Triangular and Tetrahedral Meshes
Ye, Xiu | Zhang, ShangyouCommunications on Applied Mathematics and Computation, Vol. (2024), Iss.
https://doi.org/10.1007/s42967-024-00444-4 [Citations: 0] -
Order two superconvergence of the CDG finite elements for non-self adjoint and indefinite elliptic equations
Ye, Xiu | Zhang, ShangyouAdvances in Computational Mathematics, Vol. 50 (2024), Iss. 1
https://doi.org/10.1007/s10444-023-10100-9 [Citations: 1] -
Four-order superconvergent CDG finite elements for the biharmonic equation on triangular meshes
Ye, Xiu | Zhang, ShangyouJournal of Computational and Applied Mathematics, Vol. 440 (2024), Iss. P.115516
https://doi.org/10.1016/j.cam.2023.115516 [Citations: 1] -
A superconvergent CDG finite element for the Poisson equation on polytopal meshes
Ye, Xiu | Zhang, ShangyouZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 104 (2024), Iss. 3
https://doi.org/10.1002/zamm.202300521 [Citations: 0] -
Superconvergent P1 honeycomb virtual elements and lifted P3 solutions
Lin, Yanping | Xu, Xuejun | Zhang, ShangyouCalcolo, Vol. 61 (2024), Iss. 4
https://doi.org/10.1007/s10092-024-00618-9 [Citations: 0] -
A pressure-robust stabilizer-free WG finite element method for the Stokes equations on simplicial grids
Yang, Yan | Ye, Xiu | Zhang, ShangyouElectronic Research Archive, Vol. 32 (2024), Iss. 5 P.3413
https://doi.org/10.3934/era.2024158 [Citations: 0]