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A Triangular Spectral Method for the Stokes Equations

Published online by Cambridge University Press:  28 May 2015

Lizhen Chen ,
Jie Shen  and
Chuanju Xu
Lizhen Chen
Affiliation:
School of Mathematical Sciences, Xiamen University, 361005 Xiamen, China
Jie Shen*
Affiliation:
School of Mathematical Sciences, Xiamen University, 361005 Xiamen, China Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA
Chuanju Xu*
Affiliation:
School of Mathematical Sciences, Xiamen University, 361005 Xiamen, China
*
Corresponding author.Email address:shen@math.purdue.edu
Corresponding author.Email address:cjxu@xmu.edu.cn
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Abstract

A triangular spectral method for the Stokes equations is developed in this paper. The main contributions are two-fold: First of all, a spectral method using the rational approximation is constructed and analyzed for the Stokes equations in a triangular domain. The existence and uniqueness of the solution, together with an error estimate for the velocity, are proved. Secondly, a nodal basis is constructed for the efficient implementation of the method. These new basis functions enjoy the fully tensorial product property as in a tensor-produce domain. The new triangular spectral method makes it easy to treat more complex geometries in the classical spectral-element framework, allowing us to use arbitrary triangular and tetrahedral elements.

Keywords

65N3565N2265F0535J05Stokes equationstriangular spectral methoderror analysis

Information

Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 4 , Issue 2 , May 2011 , pp. 158 - 179
Copyright
Copyright © Global Science Press Limited 2011

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References

[1]Bernardi, C. and Maday, Y.. Approximations Spectrales de Problèmes aux Limites Elliptiques. Springer-Verlag, Paris, 1992.Google Scholar
[2]Boyd, J. P.. Chebyshev and Fourier Spectral Methods. Springer-Verlag, 1989.CrossRefGoogle Scholar
[3]Braess, D. and Schwab, C.. Approximation on simplices with respect to weighted sobolev norms. J. Approx. Theory, 103(2):329–337, 2000.CrossRefGoogle Scholar
[4]Canuto, C., Hussaini, M. Y., Quarteroni, A., and Zang, T. A.. Spectral methods. Scientific Computation. Springer-Verlag, Berlin, 2006. Fundamentals in single domains.Google Scholar
[5]Dubiner, M.. Spectral methods on triangles and other domains. Journal of Scientific Computing, 6(4):345–390, 1991.CrossRefGoogle Scholar
[6]Heinrichs, W. and Loch, B. I.. Spectral schemes on triangular elements. J. Comput. Phys., 173(1):279ĺC301, 2001.CrossRefGoogle Scholar
[7]Hesthaven, J. S.. From electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex. SIAM Journal of Numerical Analysis, pages 655–676, 1998.CrossRefGoogle Scholar
[8]Karniadakis, G. E. and Sherwin, S. J.. Spectral/hp element methods for CFD. Oxford University Press, 1999.Google Scholar
[9]Koornwinder, Tom. Two-variable analogues of the classical orthogonal polynomials. In Theory and application of special functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), pages 435-495. Math. Res. Center, Univ. Wisconsin, Publ. No. 35. Academic Press, New York, 1975.Google Scholar
[10]Li, H. and Shen, J.. Optimal error estimates in Jacobi-weighted Sobolev spaces for polynomial approximations on the triangle. Math. Comp., 79:1621–1646, 2010.CrossRefGoogle Scholar
[11]Maday, Y., Meiron, D., Patera, A. T., and Rønquist, E. M.. Analysis of iterative methods for the steady and unsteady Stokes problem: Application to spectral element discretizations. SIAM Journal on Scientific Computing, 14:310, 1993.CrossRefGoogle Scholar
[12]Owens, R. G.. Spectral approximations on the triangle. Proceedings: Mathematical, Physical and Engineering Sciences, pages 857–872, 1998.CrossRefGoogle Scholar
[13]Pasquetti, R. and Rapetti, F.. Spectral element methods on triangles and quadrilaterals: comparisons and applications. Journal of Computational Physics, 198(1):349–362, 2004.CrossRefGoogle Scholar
[14]Pasquetti, R. and Rapetti, F.. Spectral element methods on unstructured meshes: Comparisons and recent advances. J. Sci. Comput., 27(1-3):377–387, 2006.CrossRefGoogle Scholar
[15]Shen, Jie, Wang, Li-Lian, and Li, Huiyuan. A triangular spectral element method using fully tensorial rational basis functions. SIAM J. Numer. Anal., 47(3):1619–1650, 2009.CrossRefGoogle Scholar
[16]Sherwin, S. J. and Karniadakis, G. E.. A triangular spectral element method: applications to the incompressible Navier-Stokes equations. Computer Methods in Applied Mechanics and Engineering, 123(1-4):189–229, 1995.CrossRefGoogle Scholar
[17]Taylor, M. A., Wingate, B. A., and Vincent, R.E.. An algorithm for computing Fekete points in the triangle. SIAM Journal on Numerical Analysis, pages 1707–1720, 2001.Google Scholar