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The vortex method with finite elements. (English) Zbl 0488.65049


MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65M25 Numerical aspects of the method of characteristics for initial value and initial-boundary value problems involving PDEs
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
76B47 Vortex flows for incompressible inviscid fluids
35Q99 Partial differential equations of mathematical physics and other areas of application
35L50 Initial-boundary value problems for first-order hyperbolic systems
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References:

[1] V. Arnold, Problèmes Ergotiques de la Mécanique Classique, Gauthier-Villars, Paris, 1967.
[2] 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. · Zbl 0399.76030
[3] 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). · Zbl 0249.35070
[4] Philippe G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. · Zbl 0383.65058
[5] Alexandre Joel Chorin, Numerical study of slightly viscous flow, J. Fluid Mech. 57 (1973), no. 4, 785 – 796.
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[8] 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.
[9] Ole Hald and Vincenza Mauceri del Prete, Convergence of vortex methods for Euler’s equations, Math. Comp. 32 (1978), no. 143, 791 – 809. , https://doi.org/10.1090/S0025-5718-1978-0492039-1 Ole H. Hald, Convergence of vortex methods for Euler’s equations. II, SIAM J. Numer. Anal. 16 (1979), no. 5, 726 – 755. · Zbl 0427.76024
[10] Ole Hald and Vincenza Mauceri del Prete, Convergence of vortex methods for Euler’s equations, Math. Comp. 32 (1978), no. 143, 791 – 809. , https://doi.org/10.1090/S0025-5718-1978-0492039-1 Ole H. Hald, Convergence of vortex methods for Euler’s equations. II, SIAM J. Numer. Anal. 16 (1979), no. 5, 726 – 755. · Zbl 0427.76024
[11] Tosio Kato, On classical solutions of the two-dimensional nonstationary Euler equation, Arch. Rational Mech. Anal. 25 (1967), 188 – 200. · Zbl 0166.45302
[12] J. A. Nitsche, \?_{\infty }-convergence of finite element approximation, Journées ”Éléments Finis”, address=Rennes, date=1975, (1975)
[13] A. C. Schaeffer, ”Existence theorem for the flow of an incompressible fluid in two-dimension,” Trans. Amer. Math. Soc., v. 42, 1967, p. 497.
[14] 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.
[15] M. Zerner, ”Equations d’évolution quasi-linéaire du premier ordre: Le cas lipschitzien.” (À paraître.) · Zbl 0481.35030
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