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Incompressible finite element methods for Navier-Stokes equations with nonstandard boundary conditions in $\textbf {R}^ 3$


Author: V. Girault
Journal: Math. Comp. 51 (1988), 55-74
MSC: Primary 65N30; Secondary 76-08, 76D05
DOI: https://doi.org/10.1090/S0025-5718-1988-0942143-2
MathSciNet review: 942143
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Abstract: This paper is devoted to the steady state, incompressible Navier-Stokes equations with nonstandard boundary conditions of the form ${\mathbf {u}} \cdot {\mathbf {n}} = 0$, $\mathbf {curl}\;{\mathbf {u}} \times {\mathbf {n}} = {\mathbf {0}}$, either on the entire boundary or mixed with the standard boundary condition ${\mathbf {u}} = {\mathbf {0}}$ on part of the boundary. The problem is expressed in terms of vector potential, vorticity and pressure. The vorticity and vector potential are approximated with curl-conforming finite elements and the pressure with standard continuous finite elements. The error estimates yield nearly optimal results for the purely nonstandard problem.


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  • Jean-Pierre Aubin, Behavior of the error of the approximate solutions of boundary value problems for linear elliptic operators by Gelerkin’s and finite difference methods, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 21 (1967), 599–637. MR 233068
  • Catherine Bègue, Carlos Conca, François Murat, and Olivier Pironneau, Ă€ nouveau sur les Ă©quations de Stokes et de Navier-Stokes avec des conditions aux limites sur la pression, C. R. Acad. Sci. Paris SĂ©r. I Math. 304 (1987), no. 1, 23–28 (French, with English summary). MR 878818
  • C. Bègue, C. Conca, F. Murat & O. Pironneau, "Les Ă©quations de Stokes et de Navier-Stokes avec des conditions aux limites sur la pression," Nonlinear Partial Differential Equations and their Applications, Collège de France Seminar (H. BrĂ©zis and J. L. Lions, eds.) (To appear.)
  • A. Bendali, J. M. DomĂ­nguez, and S. Gallic, A variational approach for the vector potential formulation of the Stokes and Navier-Stokes problems in three-dimensional domains, J. Math. Anal. Appl. 107 (1985), no. 2, 537–560. MR 787732, DOI https://doi.org/10.1016/0022-247X%2885%2990330-0
  • C. Bernardi, MĂ©thode d’ÉlĂ©ments Finis Mixtes pour les Équations de Navier-Stokes, Thèse, Univ. Paris VI, 1979.
  • Christine Bernardi, Optimal finite-element interpolation on curved domains, SIAM J. Numer. Anal. 26 (1989), no. 5, 1212–1240 (English, with French summary). MR 1014883, DOI https://doi.org/10.1137/0726068
  • 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. MR 0520174
  • Ph. ClĂ©ment, Approximation by finite element functions using local regularization, Rev. Française Automat. Informat. Recherche OpĂ©rationnelle SĂ©r. 9 (1975), no. R-2, 77–84 (English, with Loose French summary). MR 0400739
  • M. Dauge, Personal communication.
  • G. Duvaut and J.-L. Lions, Les inĂ©quations en mĂ©canique et en physique, Dunod, Paris, 1972 (French). Travaux et Recherches MathĂ©matiques, No. 21. MR 0464857
  • V. Girault, "Elementos finitos mixtos para ecuaciones de Navier-Stokes en ${{\mathbf {R}}^3}$," Acta CiĂ©nt. Venezolana. (To appear).
  • Vivette Girault and Pierre-Arnaud Raviart, Finite element methods for Navier-Stokes equations, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, Berlin, 1986. Theory and algorithms. MR 851383
  • V. Girault and P.-A. Raviart, Finite element approximation of the Navier-Stokes equations, Lecture Notes in Mathematics, vol. 749, Springer-Verlag, Berlin-New York, 1979. MR 548867
  • 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
  • J. C. NĂ©dĂ©lec, "Mixed finite elements in ${{\mathbf {R}}^3}$," Numer. Math., v. 35, 1980, pp. 315-341.
  • J.-C. NĂ©dĂ©lec, ÉlĂ©ments finis mixtes incompressibles pour l’équation de Stokes dans ${\bf R}^{3}$, Numer. Math. 39 (1982), no. 1, 97–112 (French, with English summary). MR 664539, DOI https://doi.org/10.1007/BF01399314
  • J.-C. NĂ©dĂ©lec, A new family of mixed finite elements in ${\bf R}^3$, Numer. Math. 50 (1986), no. 1, 57–81. MR 864305, DOI https://doi.org/10.1007/BF01389668
  • J. Nitsche, Ein Kriterium fĂĽr die Quasi-Optimalität des Ritzschen Verfahrens, Numer. Math. 11 (1968), 346–348 (German). MR 233502, DOI https://doi.org/10.1007/BF02166687
  • P.-A. Raviart and J. M. Thomas, A mixed finite element method for 2nd order elliptic problems, Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975) Springer, Berlin, 1977, pp. 292–315. Lecture Notes in Math., Vol. 606. MR 0483555
  • Reinhard Scholz, A mixed method for 4th order problems using linear finite elements, RAIRO Anal. NumĂ©r. 12 (1978), no. 1, 85–90, iii (English, with French summary). MR 483557, DOI https://doi.org/10.1051/m2an/1978120100851
  • R. TĂ©mam, Navier-Stokes Equations, North-Holland, Amsterdam, 1977.
  • RĂĽdiger VerfĂĽrth, Mixed finite element approximation of the vector potential, Numer. Math. 50 (1987), no. 6, 685–695. MR 884295, DOI https://doi.org/10.1007/BF01398379

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Article copyright: © Copyright 1988 American Mathematical Society