On the transport-diffusion algorithm and its applications to the Navier-Stokes equations. (English) Zbl 0505.76100

Summary: This paper deals with an algorithm for the solution of diffusion and/or convection equations where we mixed the method of characteristics and the finite element method. Globally it looks like one does one step of transport plus one step of diffusion (or projection) but the mathematics show that it is also an implicit time discretization of the PDE in Lagrangian form. We give an error bound \((h+\Delta t+h\times h/\Delta t\) in the interesting case) that holds also for the Navier-Stokes equations even when the Reynolds number is infinite (Euler equation).


76R99 Diffusion and convection
76M99 Basic methods in fluid mechanics
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65M25 Numerical aspects of the method of characteristics for initial value and initial-boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35Q30 Navier-Stokes equations
82C70 Transport processes in time-dependent statistical mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
35Q05 Euler-Poisson-Darboux equations


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[1] Bardos, C., Bercovier, M., Pironneau, O.: The Vortex Method with Finite Elements, Rapport de Recherche INRIA no 15 (1980) · Zbl 0449.76017
[2] Benque, J.P., Ibler, B., Keramsi, A., Labadie, G.: A Finite Element Method for Navier-Stokes Equations. Proceedings of the third International conference on finite elements in flow problems, Banff Alberta, Canada, 10-13 June, 1980 · Zbl 0457.76023
[3] Bercovier, M., Pironneau, O.: Error Estimates for Finite Element Method Solution of the Stokes Problem in the Primitive variables. Numerische Mathematik,33, 211-224 (1979) · Zbl 0423.65058 · doi:10.1007/BF01399555
[4] Bernadi, C.: M?thodes d’?l?ments finis mixtes pour les ?quations de Navier-Stokes, Th?se 3e cycle, Univ. Paris 6 (1979)
[5] Boris, J.P., Book, D.L.: Flux corrected transport. SHASTA J. Comput. Phys.II, 36-69 (1973) · Zbl 0251.76004
[6] Crouzeix, M., Raviart, P.A.: Conforming and non-conforming finite element methods for solving the stationary Stokes equations. RAIRO, R-3 (1973) pp. 33-76 · Zbl 0302.65087
[7] Fortin, M., Thomasset, F.: Mixed finite element methods for incompressible flow problems. J. of comp. Physics31, 113-145 (1979) · Zbl 0395.76023 · doi:10.1016/0021-9991(79)90065-2
[8] Fritts, M.J., Boris, J.P.: The Lagrangian solution of transient problems in hydronamics using a triangular mesh. J. Comp. Ph.,31, 172-215 (1979) · Zbl 0403.76033
[9] Girault, V., Raviart, P.A.: Finite Element Approximation of Navier-Stokes equations. Lecture notes in Math. Berlin G?ttingen Heidelberg: Springer 1979 · Zbl 0413.65081
[10] Glowinski, R., Mantel, B., Periaux, J., Pironneau, O.: A Finite Element Approximation of Navier-Stokes Equations for Incompressible Viscous Fluids; Computer Methods in Fluids. Morgan Taylor Brebbia ed., Pentech Press 1980 · Zbl 0426.65064
[11] Hecht, F.: Construction d’une base ? divergence nulle pour un ?l?ment fini non conforme de degr? 1 dansR 3 (to appear in RAIRO)
[12] Heywood, J., Rannacker, R.: Finite Element Approximation of the nonstationary Navier-Stokes problem (to appear)
[13] Lesaint, P., Raviart, P.A.: R?solution Num?rique de l’Equation de Continuit? par une M?thode du Type Elements Finis. Proc. Conference on Finite Elements in Rennes (1976)
[14] Pironneau, O., Raviart, P.A., Sastri, V.: Finite Element Solution of the Transport Equations by convection plus projections (to appear)
[15] Tabata, M.: A finite element approximation corresponding to the upwind differencing. Memoirs of Numerical Mathematics.1, 47-63 (1977) · Zbl 0358.65102
[16] Thomasset, F.: Finite Element Methods for Navier-Stokes Equations. VKI Lecture Series, March 25-29, 1980
[17] Zienckiewicz, O., Heinrich, J.: The finite element method and convection problem in fluid mechanics. Finite Elements in Fluids (vol. 3), Gallagher ed., New York: Wiley 1978
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