Superconvergence of Ritz-Galerkin Finite Element Approximations for Second Order Elliptic Problems

TL;DR

This paper establishes an $O(h^4)$-superconvergence rate for linear finite element solutions of second order elliptic problems, under specific geometric and coefficient conditions, supported by numerical experiments.

Contribution

It introduces an $A$-equilateral assumption that guarantees superconvergence of finite element solutions at nodal points for elliptic equations.

Findings

Superconvergence rate of $O(h^4)$ proven for specific conditions.

Numerical experiments confirm theoretical superconvergence.

Examples provided for coefficient tensors and mesh conditions.

Abstract

In this paper, the author derives an $O(h^4)$-superconvergence for the piecewise linear Ritz-Galerkin finite element approximations for the second order elliptic equation $-\nabla \cdot(A\nabla u)= f$ equipped with Dirichlet boundary conditions. This superconvergence error estimate is established between the finite element solution and the usual Lagrange nodal point interpolation of the exact solution, and thus the superconvergence at the nodal points of each element. The result is based on a condition for the finite element partition characterized by the coefficient tensor $A$ and the usual shape functions on each element, called $A$-equilateral assumption in this paper. Several examples are presented for the coefficient tensor $A$ and finite element triangulations which satisfy the conditions necessary for superconvergence. Some numerical experiments are conducted to confirm this new theory of superconvergence.