×

zbMATH — the first resource for mathematics

A nonconforming finite element method of upstream type applied to the stationary Navier-Stokes equation. (English) Zbl 0681.76032
Summary: We present a nonconforming finite element method with an upstream discretization of the convective terms for solving the stationary Navier- Stokes equations. The existence of at least one solution of the discrete problem and the convergence of subsequences of such solutions to a solution of the Navier-Stokes equations are established. In addition, under certain assumptions on the data, uniqueness of the solutions can be guaranteed and error estimates of the approximate solution are given. Moreover, some favourable properties of the discrete algebraic system are discussed.

MSC:
76D05 Navier-Stokes equations for incompressible viscous fluids
35Q30 Navier-Stokes equations
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] P. G. CIARLET, The finite element method for elliptic problems, North-Holland-Publ. Comp., Amsterdam/New York, 1978. Zbl0383.65058 MR520174 · Zbl 0383.65058
[2] [2] M. CROUZEIX, P. A. RAVIART, Conforming and Nonconforming Finite Element Methods for Solving the Stationary Stokes Equations RAIRO Numer. Anal. 3 (1973), 33-76. Zbl0302.65087 MR343661 · Zbl 0302.65087
[3] M. FORTIN, Résolution numérique des équations de Navier-Stokes par des méthodes d’éléments finis de type mixte, Proc. 2 Int. Symp. Finite Elements in Flow Problems, S. Margherita Ligure, Italy (1978).
[4] V. GIRAULT, P.-A. RAVIART, Finite Element Approximation of the Navier-Stokes Equations, Lect. Notes in Math., vol. 749, Springer Verlag, Berlin, Heidelberg, New York 1981. Zbl0441.65081 MR548867 · Zbl 0441.65081
[5] V. GIRAULT, P. A. RAVIART, An analysis of upwind schemes for the Navier-Stokes equations, SIAM J. Numer. Anal. 19 (1982) 2, 312-333. Zbl0487.76036 MR650053 · Zbl 0487.76036
[6] J. HEYWOOD, R. RANNACHER, Finite element approximation of the nonstationary Navier-Stokes problem I. Regularity of solutions and second order estimates for spatial discretization, SIAM J. Numer. Anal. 19 (1982) 2, 275-311. Zbl0487.76035 MR650052 · Zbl 0487.76035
[7] P. JAMET, P. A. RAVIART, Numerical solution of the stationary Navier-Stokes equations by finite element methods, Computing Methods in Applied Sciences and Engineering, Part 1, Lecture Notes in Computer Sciences 10 (1974), Springer Verlag. Zbl0285.76007 MR448951 · Zbl 0285.76007
[8] P. LESAINT, P. A. RAVIART, On a finite element method for solving the Neutron transport equation, in : Mathematical Aspects of Finite Elements in Partial Differential Equations (ed. by C. de Boor), Academic press, 1974. Zbl0341.65076 · Zbl 0341.65076
[9] [9] K. OHMORI, T. USHIJIMA, A Technique of Upstream Type Applied to a Linear Nonconforming Finite Element Approximation of Convective Diffusion Equations, RAIRO Numer. Anal. 18 (1984), 309-332. Zbl0586.65080 MR751761 · Zbl 0586.65080
[10] F. SCHIEWECK, L. TOBISKA, Eine upwind FEM zur Loesung des stationaeren Navier-Stokes-Problems. WZ TU Magdeburg 31 (1987) 5, 73-76. Zbl0638.76032 MR951104 · Zbl 0638.76032
[11] R. TEMAM, Navier-Stokes Equations. Theory and Numerical Analysis, North. Holland Publ. 1979. Zbl0426.35003 MR603444 · Zbl 0426.35003
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.