The vortex method with finite elements
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- by Claude Bardos, Michel Bercovier and Olivier Pironneau PDF
- Math. Comp. 36 (1981), 119-136 Request permission
Abstract:
This work shows that the method of charcteristics is well suited for the numerical solution of first order hyperbolic partial differential equations whose coefficients are approximated by functions piecewise constant on a finite element triangulation of the domain of integration. We apply this method to the numerical solution of Euler’s equation and prove convergence when the time step and the mesh size tend to zero. The proof is based upon the results of regularity given by Kato and Wolibner and on ${L^\infty }$ estimates for the solution of the Dirichlet problem given by Nitsche. The method obtained belongs to the family of vortex methods usually studied in a finite difference context.References
-
V. Arnold, Problèmes Ergotiques de la Mécanique Classique, Gauthier-Villars, Paris, 1967.
- Gregory R. Baker, The “cloud in cell” technique applied to the roll up of vortex sheets, J. Comput. Phys. 31 (1979), no. 1, 76–95. MR 531125, DOI 10.1016/0021-9991(79)90063-9
- C. Bardos, Existence et unicité de la solution de l’équation d’Euler en dimension deux, J. Math. Anal. Appl. 40 (1972), 769–790 (French). MR 333488, DOI 10.1016/0022-247X(72)90019-4
- Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174
- Alexandre Joel Chorin, Numerical study of slightly viscous flow, J. Fluid Mech. 57 (1973), no. 4, 785–796. MR 395483, DOI 10.1017/S0022112073002016 J. P. Christiansen, "Numerical solution of hydrodynamics by the method of point vortices," J. Comput. Phys., v. 13, 1973, pp. 863-879. B. Couet, O. Buneman & A. Leonard, Three Dimensional Simulation of the Free Shear Layer Using the Vortex in Cell Method (Proc. 2nd Sympos. on Turbulent Shear Flows, London, 1979).
- Pierre Grisvard, Behavior of the solutions of an elliptic boundary value problem in a polygonal or polyhedral domain, Numerical solution of partial differential equations, III (Proc. Third Sympos. (SYNSPADE), Univ. Maryland, College Park, Md., 1975) Academic Press, New York, 1976, pp. 207–274. MR 0466912
- Ole Hald and Vincenza Mauceri del Prete, Convergence of vortex methods for Euler’s equations, Math. Comp. 32 (1978), no. 143, 791–809. MR 492039, DOI 10.1090/S0025-5718-1978-0492039-1
- Ole Hald and Vincenza Mauceri del Prete, Convergence of vortex methods for Euler’s equations, Math. Comp. 32 (1978), no. 143, 791–809. MR 492039, DOI 10.1090/S0025-5718-1978-0492039-1
- Tosio Kato, On classical solutions of the two-dimensional nonstationary Euler equation, Arch. Rational Mech. Anal. 25 (1967), 188–200. MR 211057, DOI 10.1007/BF00251588
- J. A. Nitsche, $L_{\infty }$-convergence of finite element approximation, Journées “Éléments Finis” (Rennes, 1975) Univ. Rennes, Rennes, 1975, pp. 18. MR 568857 A. C. Schaeffer, "Existence theorem for the flow of an incompressible fluid in two-dimension," Trans. Amer. Math. Soc., v. 42, 1967, p. 497. W. Wolibner, "Un théorème sur l’existence du mouvement plan d’un fluide parfait homogène et incompressible pendant un temps infiniment long," Math. Z., v. 37, 1935, pp. 727-738. M. Zerner, "Equations d’évolution quasi-linéaire du premier ordre: Le cas lipschitzien." (À paraître.)
Additional Information
- © Copyright 1981 American Mathematical Society
- Journal: Math. Comp. 36 (1981), 119-136
- MSC: Primary 65N30; Secondary 65M25, 76C05
- DOI: https://doi.org/10.1090/S0025-5718-1981-0595046-3
- MathSciNet review: 595046