Superconvergence of Numerical Gradient for Weak Galerkin Finite Element Methods on Nonuniform Cartesian Partitions in Three Dimensions
This work extends a known 2D superconvergence result to 3D for weak Galerkin methods, addressing the loss of symmetry in higher dimensions.
The authors derive a superconvergence error estimate for the gradient approximation of second-order elliptic problems using weak Galerkin finite elements on nonuniform cubic partitions in 3D, achieving an O(h^r) (1.5 ≤ r ≤ 2) superconvergence order. Numerical experiments confirm the theory.
A superconvergence error estimate for the gradient approximation of the second order elliptic problem in three dimensions is analyzed by using weak Galerkin finite element scheme on the uniform and non-uniform cubic partitions. Due to the loss of the symmetric property from two dimensions to three dimensions, this superconvergence result in three dimensions is not a trivial extension of the recent superconvergence result in two dimensions \cite{sup_LWW2018} from rectangular partitions to cubic partitions. The error estimate for the numerical gradient in the $L^{2}$-norm arrives at a superconvergence order of ${\cal O}(h^r) (1.5 \leq r\leq 2)$ when the lowest order weak Galerkin finite elements consisting of piecewise linear polynomials in the interior of the elements and piecewise constants on the faces of the elements are employed. A series of numerical experiments are illustrated to confirm the established superconvergence theory in three dimensions.