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Warburton, T.; Pavarino, L. F.; Hesthaven, J. S.

A pseudo-spectral scheme for the incompressible Navier-Stokes equations using unstructured nodal elements. (English) Zbl 0961.76063

J. Comput. Phys. 164, No. 1, 1-21 (2000).
Summary: We propose a pseudo-spectral scheme for solving incompressible Navier-Stokes equations by using unstructured nodal triangles. Efficient algorithms are developed with optimal rates of convergence. Navier-Stokes simulations of Kovasznay flow, shear layer flow, and flow past a cylinder demonstrate different nodal sets considered and an alternative modal approach.

Cited in 22 Documents

MSC:

76M22 Spectral methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids

Keywords:

flow past cylinder; domain decomposition; Kovasznay flow; pseudo-spectral scheme; incompressible Navier-Stokes equations; unstructured nodal triangles; optimal rates of convergence; shear layer flow

Software:

Triangle
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Full Text: DOI

References:

[1] Bos, L., Bounding the Lebesgue function for Lagrange interpolation in a simplex, J. Approx. Theor., 38, 43 (1983) · Zbl 0546.41003
[2] Bos, L., On certain configurations in \(\textbf{R}^n\) which are unisolvent for polynomial interpolation, J. Approx. Theor., 64, 271 (1991) · Zbl 0737.41002
[3] Brown, D. L.; Minion, M. L., Performance of under-resolved two-dimensional incompressible flow simulations, J. Comput. Phys., 121 (1995) · Zbl 0849.76043
[4] Canuto, C., Stabilization of spectral methods by finite element bubble functions, Comp. Meth. Appl. Mech. Eng., 116, 13 (1994) · Zbl 0826.76056
[5] Casarin, M., Quasi-optimal Schwarz methods for the conforming spectral element discretization, SIAM J. Numer. Anal., 34, 2482 (1997) · Zbl 0889.65123
[6] Chen, Q.; Babus̆ka, I., Approximate optimal points for polynomial interpolation of real functions in an interval and in a triangle, Comp. Meth. Appl. Mech. Eng., 128, 405 (1995) · Zbl 0862.65006
[7] Chen, Q.; Babus̆ka, I., The optimal symmetrical points for polynomial interpolation of real functions in the tetrahedron, Comp. Meth. Appl. Mech. Eng., 137, 89 (1996) · Zbl 0877.65004
[8] Deville, M.; Mund, E., Finite element preconditioning for pseudospectral solutions of elliptic problems, SIAM J. Sci. Stat. Comp., 11, 311 (1990) · Zbl 0701.65075
[9] Dubiner, M., Spectral methods on triangles and other domains, J. Sci. Comp., 6, 345 (1991) · Zbl 0742.76059
[10] Fischer, P. F., An overlapping Schwarz method for spectral element solution of the incompressible Navier-Stokes equations, J. Comput. Phys., 133, 84 (1997) · Zbl 0904.76057
[11] Gordon, W. J.; Hall, C. A., Transfinite element methods: Blending function interpolation over arbitrary curved element domains, Numer. Math., 21 (1973) · Zbl 0254.65072
[12] Heinrichs, W., Spectral collocation on triangular elements, J. Comput. Phys., 145, 743 (1998) · Zbl 0909.65089
[13] Henderson, R. D., Unstructured Spectral Element Methods: Parallel Algorithms and Simulations (1994)
[14] Henderson, R. D., Details of the drag curve near the onset of vortex shedding, Phys. Fluids, 7, 1 (1995)
[15] Hesthaven, J. S., From electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex, SIAM J. Numer. Anal., 35, 655 (1998) · Zbl 0933.41004
[16] J. S. Hesthaven, and, C. H. Teng, Stable spectral methods on tetrahedral elements, SIAM J. Sci. Comp, in press.; J. S. Hesthaven, and, C. H. Teng, Stable spectral methods on tetrahedral elements, SIAM J. Sci. Comp, in press. · Zbl 0959.65112
[17] Karniadakis, G.; Israeli, M.; Orszag, S. A., High-order splitting methods for the incompressible Navier-Stokes equations, J. Comput. Phys., 97, 414 (1991) · Zbl 0738.76050
[18] Karniadakis, G.; Sherwin, S. J., Spectral hp Element Methods for CFD (1999) · Zbl 0954.76001
[19] T. Koornwinder, Two-variable analogues of the classical orthogonal polynomials, in, Theory and Applications of Special Functions, edited by, A. R. A. Academic Press, San Diego, 1975.; T. Koornwinder, Two-variable analogues of the classical orthogonal polynomials, in, Theory and Applications of Special Functions, edited by, A. R. A. Academic Press, San Diego, 1975. · Zbl 0326.33002
[20] Kovasznay, L., Laminar flow behind a two-dimensional grid, Proc. Cambridge Philos. Soc., 44 (1948) · Zbl 0030.22902
[21] Meijerink, J. A.; Van der Vorst, H. A., Guidelines for the usage of incomplete decompositions in solving sets of linear equations as they appear in practical problems, J. Comput. Phys., 44, 134 (1981) · Zbl 0472.65028
[22] Orzag, S. A., Spectral methods for problems in complex geometries, J. Comput. Phys., 70 (1980) · Zbl 0476.65078
[23] Owens, R., Spectral approximations on the triangle, Proc. R. Soc. London A, 454, 857 (1998) · Zbl 0915.35077
[24] Parter, S. V.; Rothman, E. E., Preconditioning Legendre spectral collocation approximations to elliptic problems, SIAM J. Numer. Anal., 333 (1995) · Zbl 0822.65093
[25] Pavarino, L. F.; Warburton, T., Overlapping Schwarz Methods for Unstructured Spectral Elements (1999) · Zbl 0957.65104
[26] Proriol, J., Sur une famille de polynomes á deux variables orthogonaux dans un triangle, C.R. Acad. Sci. Paris, 257, 2459 (1967) · Zbl 0080.05204
[27] Sherwin, S.; Karniadakis, G., Tetrahedral hp finite elements: Algorithms and flow simulations, J. Comput. Phys., 124, 14 (1996) · Zbl 0847.76038
[28] S. Sherwin, T. Warburton, and G. Karniadakis, Spectral/hp methods for elliptic problems on hybrid grids, in Domain Decomposition Methods 10, edited by J. Mandel, C. Farhat, and X.-C. Cai, Vol. 218, AMS, Contemporary Mathematics 1998, pp. 191-216.; S. Sherwin, T. Warburton, and G. Karniadakis, Spectral/hp methods for elliptic problems on hybrid grids, in Domain Decomposition Methods 10, edited by J. Mandel, C. Farhat, and X.-C. Cai, Vol. 218, AMS, Contemporary Mathematics 1998, pp. 191-216. · Zbl 0909.65087
[29] Sherwin, S. J., Triangular and Tetrahedral Spectral/hp Finite Element Methods for Fluid Dynamics (1995)
[30] J. R. Shewchuk, Triangle: Engineering a 2D quality mesh generator and Delaunay triangulator, in Applied Computational Geometry: Towards Geometric Engineering, edited by M. C. Lin and D. Manocha, Vol. 1148, Lecture Notes in Computer Science Springer-Verlag, New York, 1996, pp. 203-222. From the First ACM Workshop on Applied Computational Geometry.; J. R. Shewchuk, Triangle: Engineering a 2D quality mesh generator and Delaunay triangulator, in Applied Computational Geometry: Towards Geometric Engineering, edited by M. C. Lin and D. Manocha, Vol. 1148, Lecture Notes in Computer Science Springer-Verlag, New York, 1996, pp. 203-222. From the First ACM Workshop on Applied Computational Geometry.
[31] K. Stuben and U. Trottenberg, Multigrid methods: Fundamental algorithms, model problem analysis and applications, in Multigrid Methods, edited by W. Hackbusch and U. TrottenbergSpringer Berlin, New York, 1982, pp. 1-176.; K. Stuben and U. Trottenberg, Multigrid methods: Fundamental algorithms, model problem analysis and applications, in Multigrid Methods, edited by W. Hackbusch and U. TrottenbergSpringer Berlin, New York, 1982, pp. 1-176. · Zbl 0562.65071
[32] Szegö, G., Orthogonal polynomials, Colloquium Publications 23 (1939) · JFM 65.0278.03
[33] M. Taylor, and, B. Wingate, Fekete collocation points for triangular spectral elements, submitted for publication.; M. Taylor, and, B. Wingate, Fekete collocation points for triangular spectral elements, submitted for publication.
[34] Warburton, T. C.E., Spectral/hp Methods on Polymorphic Multi-Domains: Algorithms and Applications (1999)
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