A $$C^0$$ C 0 Linear Finite Element Method for Biharmonic Problems

A $$C^0$$ C 0 Linear Finite Element Method for Biharmonic Problems In this paper, a $$C^0$$ 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 $$C^0$$ 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 $$L^2$$ L 2 and discrete $$H^2$$ 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 $$C^0$$ C 0 interior penalty method is given. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Scientific Computing Springer Journals

A $$C^0$$ C 0 Linear Finite Element Method for Biharmonic Problems

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Publisher
Springer US
Copyright
Copyright © 2017 by Springer Science+Business Media, LLC
Subject
Mathematics; Algorithms; Computational Mathematics and Numerical Analysis; Mathematical and Computational Engineering; Theoretical, Mathematical and Computational Physics
ISSN
0885-7474
eISSN
1573-7691
D.O.I.
10.1007/s10915-017-0501-0
Publisher site
See Article on Publisher Site

Abstract

In this paper, a $$C^0$$ 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 $$C^0$$ 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 $$L^2$$ L 2 and discrete $$H^2$$ 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 $$C^0$$ C 0 interior penalty method is given.

Journal

Journal of Scientific ComputingSpringer Journals

Published: Jul 20, 2017

References

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