# zbMATH — the first resource for mathematics

Decentrage et elements finis mixtes pour les équations de diffusion- convection. (French) Zbl 0562.65077
The diffusion-convection equation in two spatial dimensions is approximated by mixed finite elements using an unsymmetric approach (due to Lesaint/Raviart) for the convection term: For a dissipation parameter $$0\leq \delta \leq 1$$, one gets a centered scheme in case $$\delta =0$$ and an upwind scheme if $$\delta =1$$. (Besides this, there is no indication how to choice that parameter in a practical problem, say in dependence on coefficients and triangulation.) Existence, uniqueness and error estimates are proven in the elliptic and (for the semi-discrete approximation) in the parabolic case, including estimates for diffusion tending to zero.
Reviewer: G.Stoyan

##### MSC:
 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N15 Error bounds for boundary value problems involving PDEs 35K20 Initial-boundary value problems for second-order parabolic equations
Full Text:
##### References:
 [1] P. A. Raviart, J. M. Thomas,A Mixed Finite Element Method for 2 nd Order Elliptic Problems, Mathematical Aspects of the Finite Element Method, Ed. Galligani and Magenes, Lecture Notes in Mathematics, 606, Springer, (1977). · Zbl 0362.65089 [2] P. Lesaint, P. A. Raviart,On a Finite Element Method for Solving the Neutron Transport Equation, Mathematical Aspects of Finite Elements in Partial Differential Equations, Ed. Carl de Boor, Academic Press, (1974). [3] J. Jaffre,Approximation par une méthode d’éléments finis mixtes d’une équation du type diffusion-convection linéaire staticnnaire, Rapport INRIAo 367, INRIA-Le Chesnay, France (1979). [4] P. Joly,La methode des éléments finis mixtes appliquée au problème de diffusionconvection, Thèse de 3ème cycle, Université Pierre et Marie Curie, Paris, (1982). [5] C. Johnson, V. Thomee Error Estimates for Some Mixed Finite Element Methods for Parabolic Type Problems, RAIRO, Anal. Numér.,15, (1981), 1–78. [6] J. Jaffre,Formulation mixte d’écoulements diphasiques incompressibles dans un milieu poreux Rapport INRIA no 37, INRIA-Le Chesnay, France, (1980). [7] G. Chavent et al.,Simulation of Two Dimensional Waterflooding Using Mixed Finite Elements, 6th SPE Symposium on Reservoir Simulation, New Orleans, SPE 10502, (1982). [8] J. Douglas, J. Roberts,Mixed Finite Element Methods for 2nd Order Elliptic Problems, Math. Aplic. Comput. (1),1, (1982), 91–103. · Zbl 0482.65057 [9] F. Brezzi,On the Existence, Uniqueness and Approximation of Saddle-Point Problems Arising from Lagrangian Multipliers, RAIRO, Anal. Numér..,2 (1974), 129–151. · Zbl 0338.90047 [10] P. Ciarlet,The Finite Element Method for Elliptic Problems, North-Holland, (1978). · Zbl 0383.65058 [11] P. Ciarlet, P. A. Raviart,General Lagrange and Hermite Interpolation in R n with Applications to Finite Elements Methods, Arch. Rational Mech. Anal.,46, (1972), 177–199. · Zbl 0243.41004 [12] M. Fortin,Résolution numérique des équations de Navier-Stokes par des éléments finis du type mixte, Rapport INRIA no 184, INRIA-Le Chesnay, France, (1976). [13] J. L. Lions, E. Magenes,Problèmes aux limites non homogènes, Vol. 2, Dunod, (1968). · Zbl 0165.10801 [14] J. Douglas, T. Dupont,Interior penalty procedures for elliptic and parabolic Galerkin methods, Compting Methods in Applied Science, Lecture Notes in Physics 58, Springer Verlag, (1976).
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.