Communications in Mathematical Sciences

Volume 22 (2024)

Number 1

A new priori error estimation of nonconforming element for two-dimensional linearly elastic shallow shell equations

Pages: 167 – 179

DOI: https://dx.doi.org/10.4310/CMS.2024.v22.n1.a7

Authors

Rongfang Wu (School of Computer Science and Engineering, Xi’an University of Technology, Xi’an, China)

Xiaoqin Shen (School of Computer Science and Engineering, Xi’an University of Technology, Xi’an, China)

Qian Yang (School of Computer Science and Engineering, Xi’an University of Technology, Xi’an, China)

Shengfeng Zhu (School of Mathematical Sciences, East China Normal University, Shanghai, China)

Abstract

In this paper, we mainly propose a new priori error estimation for the two-dimensional linearly elastic shallow shell equations, which rely on a family of Kirchhoff–Love theories. As the displacement components of the points on the middle surface have different regularities, the nonconforming element for the discretization shallow shell equations is analysed. Then, relying on the enriching operator, a new error estimate of energy norm is given under the regularity assumption $\vec{\zeta}_H \times \zeta_3 \in (H^{1+m} (\omega))^2 \times H^{2+m} (\omega)$ with any $m \gt 0$. Compared with the classic error analysis in other shell literature, convergence order of numerical solution can be controlled by its corresponding approximation error with an arbitrarily high order term, which fills the gap in the computational shell theory. Finally, numerical results for the saddle shell and cylindrical shell confirm the theoretical prediction.

Keywords

nonconforming elements, enriching operator, error estimation

2010 Mathematics Subject Classification

65N15, 65N30

Received 19 July 2022

Received revised 5 January 2023

Accepted 19 May 2023

Published 7 December 2023