×

zbMATH — the first resource for mathematics

Bubble stabilization of spectral Legendre methods for the advection-diffusion equation. (English) Zbl 0847.76059
Summary: An advection-diffusion model problem is discretized by a spectral Legendre method augmented by trial/test functions with local support (bubbles). The numerical analysis shows that the correction is equivalent to adding a spectrally accurate streamline-upwind stabilization term. The performances of two strategies for tuning the bubble stabilization are investigated.

MSC:
76M25 Other numerical methods (fluid mechanics) (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Canuto, C.; Hussaini, M.Y.; Quarteroni, A.; Zang, T.A., Spectral methods in fluid dynamics, (1990), Springer-Verlag Berlin New York, 2nd printing.
[2] Gottlieb, D., The Gibbs phenomenon and spectral methods, ()
[3] Tadmor, E., Convergence of spectral methods for nonlinear conservation laws, SIAM J. numer. anal., 26, 30-44, (1989) · Zbl 0667.65079
[4] Bayliss, A.; Gottlieb, D.; Matkowsky, B.J.; Minkoff, M., An actaptive pseudo-spectral method for reaction-diffusion problems, J. comput. phys., 81, 421-443, (1989) · Zbl 0668.65092
[5] Funaro, D., A new scheme for the approximation of advection-diffusion equations by collocation, SIAM J. numer. anal., 30, 1664-1676, (1993) · Zbl 0796.65120
[6] Canuto, C., Stabilization of spectral methods by finite element bubble functions, (), Comput. methods appl. mech. engrg., 116, 13-26, (1994), also · Zbl 0826.76056
[7] Arnold, D.; Brezzi, F.; Fortin, M., A stable finite element for the Stokes equations, Calcolo, 21, 337-344, (1984) · Zbl 0593.76039
[8] Brezzi, F.; Bristeau, M.-O.; Franca, L.P.; Mallet, M.; RogĂ©, G., A relationship between stabilized finite element methods and the Galerkin method with bubble functions, Comput. methods. appl. mech. engrg., 96, 117-130, (1992) · Zbl 0756.76044
[9] Baiocchi, C.; Brezzi, F.; Franca, L.P., Virtual bubbles and Galerkin-least-squares type methods, Comput. methods appl. mech. engrg., 105, 125-142, (1993) · Zbl 0772.76033
[10] Brooks, A.N.T.; Hughes, T.J.R., Streamline upwind/Petrov-Galerkin formulations for convective dominated flows with a particular emphasis on the incompressible Navier-Stokes equations, Comput. methods appl. mech. engrg., 32, 199-259, (1982) · Zbl 0497.76041
[11] Pasquarelli, F.; Quarteroni, A., Effective spectral approximations to convection-diffusion equations, (), Comput. methods appl. mech. engrg., 116, 39-51, (1994), also · Zbl 0826.76072
[12] Canuto, C., Spectral methods and a maximum principle, Math. comp., 51, 615-629, (1988) · Zbl 0699.65080
[13] Szego, G., Orthogonal polynomials, (1978), Amer. Math. Soc Providence, RI · JFM 65.0278.03
[14] Bernardi, C.; Maday, Y., Polynomial interpolation results in Sobolev spaces, J. comput. appl. math., 43, 53-80, (1992) · Zbl 0767.41001
[15] Franca, L.P.; Frey, S.L.; Hughes, J.T.R., Stabilized finite element methods: I. application to the advective-diffusive model, Comput. methods appl. mech. engrg., 92, 253-276, (1992) · Zbl 0759.76040
[16] A. Russo, personal communication.
[17] Puppo, G., Bubble stabilization of spectral methods: the multi-dimensional case, ()
[18] C. Canuto and V. van Kemenade, Bubble-stabilized spectral methods for the incompressible Navier-Stokes equations, in preparation. · Zbl 0894.76057
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.