(untitled) (Advances in Applied Mathematics and Mechanics, 2024)
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.