TY - JOUR
T1 - An Explicit Superconvergent Weak Galerkin Finite Element Method for the Heat Equation
AU - Gao , Fuzheng
AU - Zhang , Shangyou
JO - Advances in Applied Mathematics and Mechanics
VL - 2
SP - 538
EP - 553
PY - 2024
DA - 2024/12
SN - 17
DO - http://doi.org/10.4208/aamm.OA-2023-0290
UR - https://global-sci.org/intro/article_detail/aamm/23733.html
KW - Parabolic equations, finite element, weak Galerkin method, polytopal mesh.
AB - <p style="text-align: justify;">An explicit two-order superconvergent weak Galerkin finite element
method is designed and analyzed for the heat equation on triangular and tetrahedral grids. For two-order superconvergent $P_k$ weak Galerkin finite elements, the auxiliary inter-element functions must be $P_{k+1}$ polynomials. In order to achieve the superconvergence, the usual $H^1$-stabilizer must be also eliminated. For time-explicit weak
Galerkin method, a time-stabilizer is added, on which the time-derivative of the auxiliary variables can be defined. But for the two-order superconvergent weak Galerkin
finite elements, the time-stabilizer must be very weak, an $H^{−2}$-like inner-product instead of an $L^2$-like inner-product. We show the two-order superconvergence for both
semi-discrete and fully-discrete schemes. Numerical examples are provided.</p>