Adv. Appl. Math. Mech., 17 (2025), pp. 538-553.
Published online: 2024-12
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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.
}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2023-0290}, url = {http://global-sci.org/intro/article_detail/aamm/23733.html} }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.