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20-07-2017 | Issue 3/2018

Journal of Scientific Computing 3/2018

A \(C^0\) Linear Finite Element Method for Biharmonic Problems

Journal:
Journal of Scientific Computing > Issue 3/2018
Authors:
Hailong Guo, Zhimin Zhang, Qingsong Zou
Important notes
H. Guo: The research of this author was supported in part by the US National Science Foundation through Grant DMS-1419040. Z. Zhang: The research of this author was supported in part by the following Grants: NSFC 11471031, NSFC 91430216, NASF U1530401, and NSF DMS-1419040. Q. Zou: The research of this author was supported in part by the following Grants: the special project High performance computing of National Key Research and Development Program 2016YFB0200604, NSFC 11571384, Guangdong Provincial NSF 2014A030313179, the Fundamental Research Funds for the Central Universities 16lgjc80.

Abstract

In this paper, a \(C^0\) linear finite element method for biharmonic equations is constructed and analyzed. In our construction, the popular post-processing gradient recovery operators are used to calculate approximately the second order partial derivatives of a \(C^0\) linear finite element function which do not exist in traditional meaning. The proposed scheme is straightforward and simple. More importantly, it is shown that the numerical solution of the proposed method converges to the exact one with optimal orders both under \(L^2\) and discrete \(H^2\) norms, while the recovered numerical gradient converges to the exact one with a superconvergence order. Some novel properties of gradient recovery operators are discovered in the analysis of our method. In several numerical experiments, our theoretical findings are verified and a comparison of the proposed method with the nonconforming Morley element and \(C^0\) interior penalty method is given.

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