An efficient Legendre–Galerkin spectral element method for the steady flows in rectangular cavities

Abstract

An efficient Legendre–Galerkin spectral element method for the steady flows in rectangular cavities is proposed in this paper. Firstly, we eliminate the singularity of biharmonic equation in rectangular cavity at the corner by the singularity substraction technique. Then we construct some appropriate interior basis functions and interface basis functions which maintain $C^1$-continuity. Consequently, the discrete variational formulation is reduced to a linear system with block diagonal and well-conditioned coefficient matrix, which can be efficiently solved by the conjugate gradient iteration method. Finally, several numerical examples are given to show the effectiveness of our numerical method. The present method is used to solve the creeping flows in rectangular cavities, the numerical results are compared well with the benchmark steady solutions provided by the finite difference method.

Keywords:

• Legendre–Galerkin
• spectral element method
• domain decomposition
• rectangular cavities
• conjugate gradient iteration

2010 AMS SUBJECT CLASSIFICATIONS:

• 65N35
• 65N55
• 76B10

Acknowledgments

We would like to thank the anonymous referees for their valuable suggestions and comments, which greatly helped improve the presentation of this manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work of Jun Zhang is supported by the National Natural Science Foundation of China (no.11901132), the Chinese Postdoc Foundation Grant (no. 2019M653490), and the Academic Project of Guizhou University of Finance and Economics (no. [2018]5774-033). The work of Jianjun Jiao is supported by National Natural Science Foundation of China (nos. 11791019 and 11361014), the Joint Fund Project of Department of Commerce with GUFE (no. 2016SWBZD18), the Science Technology Foundation of Guizhou Education Department (nos. QJK[KY]2018019, [2019]1051), the Project of High Level Creative Talents in Guizhou Province (no. 20164035), and the Guizhou Province University science and technology top talents project (no. 2018-047). The work of Tao Sun is supported in part by NSF of China (nos. 11401380, 11671166 and 11701371) and also by NSF of Shanghai (no. 19ZR1436300).