CSIAM Trans. Appl. Math., 4 (2023), pp. 256-274.
Published online: 2023-02
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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.
}, issn = {2708-0579}, doi = {https://doi.org/10.4208/csiam-am.SO-2021-0051}, url = {http://global-sci.org/intro/article_detail/csiam-am/21414.html} }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.