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 Li et al. (0000) from rectangular partitions to cubic partitions. The error estimate for the numerical gradient in the L2-norm arrives at a superconvergence order of O(hr)(1.5≤r≤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.
- Non-uniform cubic partitions
- Second order elliptic problem
- Three dimensions
- Weak Galerkin finite element method