The triangular spectral element method for Stokes eigenvalues
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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.References
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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
- MR Author ID: 708582
- Email: huiyuan@iscas.ac.cn
- 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).
- © Copyright 2017 American Mathematical Society
- Journal: Math. Comp. 86 (2017), 2579-2611
- MSC (2010): Primary 65N35, 65N25, 35P15, 35Q30
- DOI: https://doi.org/10.1090/mcom/3173
- MathSciNet review: 3667018