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This article presents a superconvergence for the gradient approximation ofthe second order elliptic equation discretized by the weak Galerkin finiteelement methods on nonuniform rectangular partitions. The result shows aconvergence of ${\cal O}(h^r)$, $1.5\leq r \leq 2$, for the numerical gradientobtained from the lowest order weak Galerkin element consisting of piecewiselinear and constant functions. For this numerical scheme, the optimal order oferror estimate is ${\cal O}(h)$ for the gradient approximation. Thesuperconvergence reveals a superior performance of the weak Galerkin finiteelement methods. Some computational results are included to numericallyvalidate the superconvergence theory.

Superconvergence of the Gradient Approximation for Weak Galerkin Finite Element Methods on Nonuniform Rectangular Partitions