Superconvergence of Numerical Gradient for Weak Galerkin Finite Element Methods on Nonuniform Cartesian Partitions in Three Dimensions

TL;DR

This paper establishes a superconvergence error estimate for the gradient approximation in weak Galerkin finite element methods applied to 3D elliptic problems on nonuniform cubic partitions, confirmed by numerical experiments.

Contribution

It extends superconvergence results from 2D to 3D for weak Galerkin methods on nonuniform partitions, addressing the loss of symmetry.

Findings

Superconvergence order of ${ m O}(h^r)$ with $1.5 ext{ to } 2$ for gradient approximation.

Numerical experiments confirm the theoretical superconvergence results.

Extension of superconvergence analysis to three dimensions on nonuniform cubic partitions.

Abstract

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.