Global superconvergence of the lowest order mixed finite element on mildly structured meshes

TL;DR

This paper establishes global superconvergence estimates for the lowest order Raviart--Thomas mixed finite element method on mildly structured triangular meshes, demonstrating improved accuracy through mesh structure and postprocessing techniques.

Contribution

It introduces new superconvergence results for mixed finite elements on mildly structured meshes, including a postprocessing method to enhance solution accuracy.

Findings

Superconvergence order of $1+ ho$ for the vector variable in $L^{2}$ norm.

Postprocessing operator $G_{h}$ improves accuracy to order $1+ ho$.

Superconvergence estimates also apply to Crouzeix--Raviart nonconforming elements.

Abstract

In this paper, we develop global superconvergence estimates for the lowest order Raviart--Thomas mixed finite element method for second order elliptic equations with general boundary conditions on triangular meshes, where most pairs of adjacent triangles form approximate parallelograms. In particular, we prove the $L^{2}$-distance between the numerical solution and canonical interpolant for the vector variable is of order $1+\rho$, where $\rho\in(0,1]$ is dependent on the mesh structure. By a cheap local postprocessing operator $G_{h}$, we prove the $L^{2}$-distance between the exact solution and the postprocessed numerical solution for the vector variable is of order $1+\rho$. As a byproduct, we also obtain the superconvergence estimate for Crouzeix--Raviart nonconforming finite elements on triangular meshes of the same type.