A combined spectral-finite element method for solving two-dimensional unsteady Navier-Stokes equations. (English) Zbl 0758.76032

In the paper a spectral finite element approach for solving two dimensional, semi-periodic unsteady Navier-Stokes equations is generalized. Previous methods which are mentioned in the introduction, i.e. the finite difference method or combined Fourier-Chebyshev approximation, cannot easily be extended to three dimensional problems with non rectangular domains. The advantage of the method proposed is that it can be generalized easily to more complicated problems. The solution scheme for the combined spectral FEM is constructed. The resulting equation is approximated directly while the incompressibility condition first is softened by the artificial compression term. It leads in fact to the penalty method. The penalty function is calculated with reduced integration to increase the convergence. The convergence theorem is given and proved. Numerical results exhibit lower error in the case of the proposed method compared to the finite element method.
The paper is written concisely and may be interesting for researchers as well as those who wish to develop numerical tools of calculations.


76M10 Finite element methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
Full Text: DOI


[1] Murdock, J. W., AIAA J., 15, 1167 (1977)
[2] Murdock, J. W., AIAA Paper 86-0434 (1986), (unpublished)
[3] Ingham, D. B., J. Comput. Phys., 53, 90 (1984)
[4] Ingham, D. B., (Proc. R. Soc. London A, 402 (1985)), 109
[5] Beringen, S., J. Fluid Mech., 148, 413 (1984)
[6] Milinazo, F. A.; Saffman, P. G., J. Fluid Mech., 160, 281 (1985)
[7] Guo, B.-Y., J. Comput. Math., 6, 238 (1988)
[8] Canuto, C.; Maday, Y.; Quarteroni, A., Numer. Math., 39, 205 (1982)
[9] Canuto, C.; Maday, Y.; Quarteroni, A., Numer. Math., 44, 201 (1984)
[10] Guo, B.-Y.; Cao, W.-M., Acta Math. Appl. Sin., 7, 1 (1991)
[11] Orszag, S. A., J. Fluid Mech., 49, 75 (1971)
[12] Orszag, S. A.; Kells, L. C., J. Fluid Mech., 96, 159 (1980)
[13] Rozhdestvensky, B. L.; Simakin, I. N., J. Fluid Mech., 147, 261 (1984)
[14] Téman, R., Navier-Stokes Equations (1977), North-Holland: North-Holland Amsterdam · Zbl 0406.35053
[15] Ying, L., Adv. in Math., 12, 124 (1983)
[16] Oden, J. T., RIP—Methods for Stokesian Flows, (Gallagher, R. H.; etal., Finite Element Methods in Fluids (1982), Wiley: Wiley New York) · Zbl 0447.76029
[17] Grisvard, P., Ann. Sci. Ecole Norm. Sup., 4, 311 (1969)
[18] Ciarlet, P. G., The Finite Element Method for Elliptic Problems (1978), NorthHolland: NorthHolland Amsterdam · Zbl 0445.73043
[19] Guo, B.-Y., Difference Methods for Partial Differential Equations (1988), Science Press: Science Press Beijing
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.