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Korn's inequalities for piecewise $H^1$ vector fields

Author: Susanne C. Brenner
Journal: Math. Comp. 73 (2004), 1067-1087
MSC (2000): Primary 65N30, 74S05
Published electronically: September 26, 2003
MathSciNet review: 2047078
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Abstract: Korn's inequalities for piecewise $H^1$ vector fields are established. They can be applied to classical nonconforming finite element methods, mortar methods and discontinuous Galerkin methods.

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  • 1. J. H. Bramble and S. R. Hilbert, Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation, SIAM J. Numer. Anal. 7 (1970), 112–124. MR 0263214
  • 2. S.C. Brenner.
    Poincaré-Friedrichs inequalities for piecewise $H^1$ functions.
    SIAM J. Numer. Anal. 41: 306-324, 2003 (electronic).
  • 3. Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, 2nd ed., Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 2002. MR 1894376
  • 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. MR 0520174
  • 5. Philippe G. Ciarlet, Mathematical elasticity. Vol. I, Studies in Mathematics and its Applications, vol. 20, North-Holland Publishing Co., Amsterdam, 1988. Three-dimensional elasticity. MR 936420
  • 6. Bernardo Cockburn, George E. Karniadakis, and Chi-Wang Shu (eds.), Discontinuous Galerkin methods, Lecture Notes in Computational Science and Engineering, vol. 11, Springer-Verlag, Berlin, 2000. Theory, computation and applications; Papers from the 1st International Symposium held in Newport, RI, May 24–26, 1999. MR 1842160
  • 7. J. Dieudonné, Foundations of modern analysis, Pure and Applied Mathematics, Vol. X, Academic Press, New York-London, 1960. MR 0120319
  • 8. G. Duvaut and J.-L. Lions, Inequalities in mechanics and physics, Springer-Verlag, Berlin-New York, 1976. Translated from the French by C. W. John; Grundlehren der Mathematischen Wissenschaften, 219. MR 0521262
  • 9. Richard S. Falk, Nonconforming finite element methods for the equations of linear elasticity, Math. Comp. 57 (1991), no. 196, 529–550. MR 1094947, 10.1090/S0025-5718-1991-1094947-6
  • 10. M. Fortin, A three-dimensional quadratic nonconforming element, Numer. Math. 46 (1985), no. 2, 269–279. MR 787211, 10.1007/BF01390424
  • 11. M. Fortin and M. Soulie, A nonconforming piecewise quadratic finite element on triangles, Internat. J. Numer. Methods Engrg. 19 (1983), no. 4, 505–520. MR 702056, 10.1002/nme.1620190405
  • 12. P. Hansbo and M.G. Larson.
    Discontinuous Galerkin methods for incompressible and nearly incompressible elasticity by Nitsche's method.
    Comput. Methods Appl. Mech. Engrg., 191:1895-1908, 2002.
  • 13. P. Knobloch.
    On Korn's inequality for nonconforming finite elements.
    Technische Mechanik, 20:205-214 and 375 (Errata), 2000.
  • 14. J. A. Nitsche, On Korn’s second inequality, RAIRO Anal. Numér. 15 (1981), no. 3, 237–248 (English, with French summary). MR 631678
  • 15. Christoph Schwab, ℎ𝑝-FEM for fluid flow simulation, High-order methods for computational physics, Lect. Notes Comput. Sci. Eng., vol. 9, Springer, Berlin, 1999, pp. 325–438. MR 1712280, 10.1007/978-3-662-03882-6_4
  • 16. Ming Wang, The generalized Korn inequality on nonconforming finite element spaces, Math. Numer. Sinica 16 (1994), no. 1, 108–113 (Chinese, with English summary); English transl., Chinese J. Numer. Math. Appl. 16 (1994), no. 2, 91–96. MR 1391700
  • 17. E.L. Wilson, R.L. Taylor, W. Doherty, and J. Ghaboussi.
    Incompatible displacement models.
    In S.J. Fenves, N. Perrone, A.R. Robinson, and W.C. Schnobrich, editors, Numerical and Computer Methods in Structural Mechanics, pages 43-57. Academic Press, New York, 1973.
  • 18. Barbara I. Wohlmuth, Discretization methods and iterative solvers based on domain decomposition, Lecture Notes in Computational Science and Engineering, vol. 17, Springer-Verlag, Berlin, 2001. MR 1820470
  • 19. Xuejun Xu, A discrete Korn’s inequality in two and three dimensions, Appl. Math. Lett. 13 (2000), no. 4, 99–102. MR 1752148, 10.1016/S0893-9659(99)00217-7
  • 20. Zhimin Zhang, Analysis of some quadrilateral nonconforming elements for incompressible elasticity, SIAM J. Numer. Anal. 34 (1997), no. 2, 640–663. MR 1442932, 10.1137/S0036142995282492

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Additional Information

Susanne C. Brenner
Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208

Keywords: Korn's inequalities, piecewise $H^1$ vector fields, nonconforming finite elements, mortar methods, discontinuous Galerkin methods
Received by editor(s): March 19, 2002
Received by editor(s) in revised form: December 14, 2002
Published electronically: September 26, 2003
Additional Notes: This work was supported in part by the National Science Foundation under Grant No. DMS-00-74246.
Article copyright: © Copyright 2003 American Mathematical Society