Superconvergence of both two and three dimensional rectangular Morley elements for biharmonic equations

TL;DR

This paper proves second-order superconvergence for 2D and 3D rectangular Morley elements solving biharmonic equations, using a novel analysis and postprocessing, without extra boundary conditions.

Contribution

It is the first to establish full second-order superconvergence for first-order nonconforming elements in 3D biharmonic problems without additional boundary conditions.

Findings

Superconvergence achieved after postprocessing

First-time superconvergence for 3D nonconforming elements

Numerical results confirm theoretical predictions

Abstract

In the present paper, superconvergence of second order, after an appropriate postprocessing, is achieved for both the two and three dimensional first order rectangular Morley elements of biharmonic equations. The analysis is dependent on superconvergence of second order for the consistency error and a corrected canonical interpolation operator, which help to establish supercloseness of second order for the corrected canonical interpolation. Then the final superconvergence follows a standard postprocessing. For first order nonconforming finite element methods of both two and three dimensional fourth order elliptic problems, it is the first time that full superconvergence of second order is obtained without an extra boundary condition imposed on exact solutions. It is also the first time that superconvergence is established for nonconforming finite element methods of three dimensional fourth order elliptic problems. Numerical results are presented to demonstrate the theoretical results.