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The triangular spectral element method for Stokes eigenvalues


Authors: Weikun Shan and Huiyuan Li
Journal: Math. Comp. 86 (2017), 2579-2611
MSC (2010): Primary 65N35, 65N25, 35P15, 35Q30
DOI: https://doi.org/10.1090/mcom/3173
Published electronically: March 29, 2017
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Abstract: A triangular spectral element method is proposed for Stokes eigenvalues, which utilizes the generalized orthogonal Koornwinder polynomials as the local basis functions. The local polynomial projection, which serves as a Fortin interpolation on each triangular element, is defined by the truncated Koornwinder-Fourier series. A sharp estimate on the discrete inf-sup constant of the divergence for our triangular spectral element approximation scheme is then acquired via the stability analysis of the local projection operator. Further, the optimal error estimate of the $ H^1$-orthogonal spectral element projection oriented to Stokes equations is obtained through the globally continuous piecewise polynomial assembled by the union of all local projections. In the sequel, the optimal convergence rate/error estimate theory is eventually established for our triangular spectral element method for both eigenvalue and source problems of the Stokes equations. Finally, numerical experiments are presented to illustrate our theories on both the discrete inf-sup constant of the divergence and the accuracy of the computational eigenvalues.


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Additional Information

Weikun Shan
Affiliation: Laboratory of Parallel Computing, Institute of Software, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China – and – University of Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
Email: shanweikun11@mails.ucas.ac.cn

Huiyuan Li
Affiliation: State Key Laboratory of Computer Science/Laboratory of Parallel Computing, Institute of Software, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
Email: huiyuan@iscas.ac.cn

DOI: https://doi.org/10.1090/mcom/3173
Keywords: Stokes eigenvalues, triangular spectral element method, generalized orthogonal Koornwinder polynomials, error analysis
Received by editor(s): April 12, 2015
Received by editor(s) in revised form: April 14, 2015, and February 4, 2016
Published electronically: March 29, 2017
Additional Notes: This work was supported by National Natural Science Foundation of China (No. 91130014, 11471312 and 91430216).
Article copyright: © Copyright 2017 American Mathematical Society

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