Superconvergence of the gradient approximation for weak Galerkin finite element methods on nonuniform rectangular partitions

Abstract

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 O(h r ), 1.5 ≤ r ≤ 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 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.Key words. weak Galerkin, finite element methods, second order elliptic equations, superconvergence, nonuniform rectangular partitions.

Simplified weak Galerkin and new finite difference schemes for the Stokes equation

“…As to the first two terms on the right-hand side of (6.9), it has been shown in [11] that exists another function σ b = (σ b ) t and a remainder R 5 such that…”

“…WG-FEM has the flexibility of dealing with discontinuous elements while sharing the same weak formulation with the classical conforming finite element methods. Following its first development in [18,19] for second order elliptic equations, the method has gained a lot attention and popularity and has been applied to various PDEs including the Stokes equation, biharmonic equation, elasticity equation, and Maxwell's equations, see [16,21,11] and the references therein for more details.…”

Superconvergence of numerical gradient for weak Galerkin finite element methods on nonuniform Cartesian partitions in three dimensions

“…We are in a position to review the weak Galerkin finite element method for the second order elliptic model problem (1.1) based on the weak formulation (1.2) [37,22].…”

A discrete maximum principle for the weak Galerkin finite element method on nonuniform rectangular partitions

“…From (2.3), the linear extension of u b ∈ W(T) can be represented as (see Lem. 6.1 in [9] for details)…”

“…Observe that a superconvergence theory in the H 1 ‐norm has been developed in for the diffusion equation on rectangular partitions; a slight modification of the analysis there will yield a superconvergence of order for the SWG solutions of the full convection–diffusion equation (1.1) and (1.2).…”