Superconvergence of the Gradient Approximation for Weak Galerkin Finite Element Methods on Nonuniform Rectangular Partitions
It provides a theoretical improvement for numerical gradient accuracy in weak Galerkin methods, benefiting computational scientists using these methods for elliptic equations.
The paper proves superconvergence of order O(h^r) (1.5 ≤ r ≤ 2) for gradient approximation in weak Galerkin finite element methods on nonuniform rectangular partitions, improving over the optimal O(h) estimate.
This article presents a superconvergence for the gradient approximation of the second order elliptic equation discretized by the weak Galerkin finite element methods on nonuniform rectangular partitions. The result shows a convergence of ${\cal O}(h^r)$, $1.5\leq r \leq 2$, for the numerical gradient obtained from the lowest order weak Galerkin element consisting of piecewise linear and constant functions. For this numerical scheme, the optimal order of error estimate is ${\cal O}(h)$ for the gradient approximation. The superconvergence reveals a superior performance of the weak Galerkin finite element methods. Some computational results are included to numerically validate the superconvergence theory.