TY - JOUR
T1 - Order Two Superconvergence of the CDG Finite Elements on Triangular and Tetrahedral Meshes
AU - Ye , Xiu
AU - Zhang , Shangyou
JO - CSIAM Transactions on Applied Mathematics
VL - 2
SP - 256
EP - 274
PY - 2023
DA - 2023/02
SN - 4
DO - http://doi.org/10.4208/csiam-am.SO-2021-0051
UR - https://global-sci.org/intro/article_detail/csiam-am/21414.html
KW - Finite element, conforming discontinuous Galerkin method, stabilizer free, triangular grid, tetrahedral grid.
AB - <p style="text-align: justify;">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.</p>