The Gradient Superconvergence of Bilinear Finite Volume Element for Elliptic Problems

TL;DR

This paper proves that the bilinear finite volume element method for elliptic problems exhibits gradient superconvergence, with the gradient approximation converging faster than the standard rate at specific points.

Contribution

The paper establishes a superclose weak estimate and proves gradient superconvergence of the bilinear FVE method for elliptic problems, including the convergence rate.

Findings

Gradient approximation converges at rate O(h^2)|ln h| at optimal stress points.

Superclose weak estimate for the bilinear form of the FVE method.

Gradient superconvergence occurs at mesh points, midpoints, and element centers.

Abstract

We study the gradient superconvergence of bilinear finite volume element (FVE) solving the elliptic problems. First, a superclose weak estimate is established for the bilinear form of the FVE method. Then, we prove that the gradient approximation of the FVE solution has the superconvergence property: $\max_{P\in S}|(\nabla u-\overline{\nabla}u_h)(P)|=O(h^2)|\ln h|$, where $\overline{\nabla}u_h(P)$ denotes the average gradient on elements containing point $P$ and $S$ is the set of optimal stress points composed of the mesh points, the midpoints of edges and elements.