A C0 Linear Finite Element Method for Biharmonic Problems

• Published in: Journal of scientific computing Vol. 74; no. 3; pp. 1397 - 1422
• Main Authors: Guo, Hailong, Zhang, Zhimin, Zou, Qingsong
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.03.2018; Springer Nature B.V
• Subjects: Algorithms; Biharmonic equations; Boundary conditions; Computational Mathematics and Numerical Analysis; Convergence; Finite element method; Inequality; Mathematical analysis; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Methods; Operators (mathematics); Recovery; Theoretical; Biharmonic equation; Superconvergence; Secondary 45N08; Linear finite element; Primary 65N30; Gradient recovery
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1007/s10915-017-0501-0

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.