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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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Korn’s inequalities for piecewise $H^1$ vector fields
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by Susanne C. Brenner HTML | PDF
Math. Comp. 73 (2004), 1067-1087 Request permission

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.
References
  • 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 263214, DOI 10.1137/0707006
  • S.C. Brenner. Poincaré-Friedrichs inequalities for piecewise $H^1$ functions. SIAM J. Numer. Anal. 41: 306–324, 2003 (electronic).
  • 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, DOI 10.1007/978-1-4757-3658-8
  • 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
  • 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
  • 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, DOI 10.1007/978-3-642-59721-3
  • J. Dieudonné, Foundations of modern analysis, Pure and Applied Mathematics, Vol. X, Academic Press, New York-London, 1960. MR 0120319
  • G. Duvaut and J.-L. Lions, Inequalities in mechanics and physics, Grundlehren der Mathematischen Wissenschaften, vol. 219, Springer-Verlag, Berlin-New York, 1976. Translated from the French by C. W. John. MR 0521262
  • Richard S. Falk, Nonconforming finite element methods for the equations of linear elasticity, Math. Comp. 57 (1991), no. 196, 529–550. MR 1094947, DOI 10.1090/S0025-5718-1991-1094947-6
  • M. Fortin, A three-dimensional quadratic nonconforming element, Numer. Math. 46 (1985), no. 2, 269–279. MR 787211, DOI 10.1007/BF01390424
  • 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, DOI 10.1002/nme.1620190405
  • 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.
  • P. Knobloch. On Korn’s inequality for nonconforming finite elements. Technische Mechanik, 20:205–214 and 375 (Errata), 2000.
  • J. A. Nitsche, On Korn’s second inequality, RAIRO Anal. Numér. 15 (1981), no. 3, 237–248 (English, with French summary). MR 631678
  • 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, DOI 10.1007/978-3-662-03882-6_{4}
  • 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
  • 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.
  • 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, DOI 10.1007/978-3-642-56767-4
  • Xuejun Xu, A discrete Korn’s inequality in two and three dimensions, Appl. Math. Lett. 13 (2000), no. 4, 99–102. MR 1752148, DOI 10.1016/S0893-9659(99)00217-7
  • Zhimin Zhang, Analysis of some quadrilateral nonconforming elements for incompressible elasticity, SIAM J. Numer. Anal. 34 (1997), no. 2, 640–663. MR 1442932, DOI 10.1137/S0036142995282492
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Additional Information
  • Susanne C. Brenner
  • Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
  • Email: brenner@math.sc.edu
  • 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.
  • © Copyright 2003 American Mathematical Society
  • Journal: Math. Comp. 73 (2004), 1067-1087
  • MSC (2000): Primary 65N30, 74S05
  • DOI: https://doi.org/10.1090/S0025-5718-03-01579-5
  • MathSciNet review: 2047078