The triangular spectral element method for Stokes eigenvalues

Shan, Weikun; Li, Huiyuan

doi:10.1090/mcom/3173

The triangular spectral element method for Stokes eigenvalues
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by Weikun Shan and Huiyuan Li PDF

Math. Comp. 86 (2017), 2579-2611 Request permission

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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