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Inverse inequalities on non-quasi-uniform meshes and application to the mortar element method

Authors: W. Dahmen, B. Faermann, I. G. Graham, W. Hackbusch and S. A. Sauter
Journal: Math. Comp. 73 (2004), 1107-1138
MSC (2000): Primary 65N12, 65N30, 65N38, 65N55, 41A17, 46E35
Published electronically: July 14, 2003
MathSciNet review: 2047080
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Abstract: We present a range of mesh-dependent inequalities for piecewise constant and continuous piecewise linear finite element functions $u$ defined on locally refined shape-regular (but possibly non-quasi-uniform) meshes. These inequalities involve norms of the form $\left\Vert h^{\alpha}u\right\Vert _{W^{s,p}(\Omega)}$ for positive and negative $s$ and $\alpha$, where $h$ is a function which reflects the local mesh diameter in an appropriate way. The only global parameter involved is $N$, the total number of degrees of freedom in the finite element space, and we avoid estimates involving either the global maximum or minimum mesh diameter. Our inequalities include new variants of inverse inequalities as well as trace and extension theorems. They can be used in several areas of finite element analysis to extend results--previously known only for quasi-uniform meshes--to the locally refined case. Here we describe applications to (i) the theory of nonlinear approximation and (ii) the stability of the mortar element method for locally refined meshes.

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  • 1. R.E. Bank and L.R. Scott: On the conditioning of finite element equations with highly refined meshes, SIAM J. Numer. Anal. 26 (1989) 1383-1394.MR 90m:65192
  • 2. C. Bernardi, Y. Maday, A.T. Patera: A new nonconforming approach to domain decomposition: the mortar element method, in: Nonlinear Partial Differential Equations and Their Applications, eds. H. Brezis and J.L. Lions, Pitman, New York, 1994 13-51. MR 95a:65201
  • 3. P. Binev, W. Dahmen, R. DeVore and P. Petrushev: Approximation classes for adaptive methods, Serdica Math. J., 28 (2002), 1001-1026.
  • 4. D. Braess: Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, Cambridge University Press, 1997. MR 98f:65002
  • 5. D. Braess and W. Dahmen: Stability estimates of the mortar finite element method for 3-dimensional problems, East-West J. Numer. Anal., 6 (1998) 249-264. MR 2000e:65115
  • 6. D. Braess, W. Dahmen, C. Wieners: A multigrid algorithm for the mortar finite element method, SIAM J. Numer. Anal., 37 (1999) 48-69. MR 2000i:65207
  • 7. D. Braess, M. Dryja, W. Hackbusch: A multigrid method for nonconforming FE-discretisations with application to nonmatching grids, Computing, 63 (1999) 1-25. MR 2000h:65048
  • 8. S. C. Brenner and L. R. Scott: The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York, 1994. MR 95f:65001
  • 9. F. Brezzi and M. Fortin: Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York - Berlin - Heidelberg, 1991. MR 92d:65187
  • 10. P.G. Ciarlet: The Finite Element Method for Elliptic Problems, North Holland, Amsterdam, 1978. MR 58:25001
  • 11. S. Dahlke, R. DeVore: Besov regularity for elliptic boundary value problems, Comm. Partial Differential Equations, 22 (1997) 1-16. MR 97k:35047
  • 12. W. Dahmen, B. Faermann, I.G. Graham, W. Hackbusch, S.A. Sauter, Inverse inequalities on non-quasi-uniform meshes and application to the mortar element method, Report 24/2001, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, 2001.
  • 13. R. DeVore: Nonlinear Approximation, Acta Numerica 7 (1998), Cambridge University Press, 51-150. MR 2001a:41034
  • 14. R. DeVore, V.A. Popov: Interpolation of Besov spaces, Trans. Amer. Math. Soc., 305 (1988) 397-414. MR 89h:46044
  • 15. R. DeVore, X.M. Yu: Degree of adaptive approximation, Math. Comp., 55 (1990) 625-635. MR 91g:41022
  • 16. B. Faermann: Localization of the Aronszajn-Slobodeckij norm and application to adaptive boundary element methods. Part I. The two-dimensional case, IMA J. Numer. Anal., 20 (2000) 203-234. MR 2001e:65192
  • 17. B. Faermann: Localization of the Aronszajn-Slobodeckij norm and application to adaptive boundary element methods. Part II. The three-dimensional case. Numer. Math., 92 (2002), 467-499.
  • 18. I.G. Graham, W. Hackbusch, S.A. Sauter: Discrete boundary element methods on general meshes in 3D, Numer. Math., 86 (2000) 103-137. MR 2001e:65224
  • 19. I.G. Graham, W. Hackbusch, S.A. Sauter: The hybrid Galerkin boundary element method: theory and implementation, Numer. Math., 86 (2000) 139-172. MR 2001d:65160
  • 20. W. Hackbusch: Elliptic Differential Equations. Theory and Numerical Treatment, SCM 18. Springer-Verlag, Berlin, 1992. MR 94b:35001
  • 21. K. Eriksson, D. Estep, P. Hansbo and C. Johnson: Introduction to adaptive methods for differential equations, Acta Numerica (1995) 105-158. MR 96k:65057
  • 22. C. Kim, R.D. Lazarov, J.E. Pasciak, P.S. Vassilevski: Multiplier spaces for the mortar finite element method in three dimensions, SIAM J. Numer. Analysis, 39 (2001) 519-538 MR 2002g:65143
  • 23. H. Triebel: Interpolation theory, function spaces, differential operators, 2nd edition, Johann Ambrosius Barth Verlag, Heidelberg, 1995. MR 96f:46001
  • 24. A.H. Schatz, V. Thomée and W.L. Wendland, Mathematical Theory of Finite and Boundary Elements, Birkhäuser-Verlag, Basel, 1990. MR 92f:65004
  • 25. B. Wohlmuth: Hierarchical a posteriori error estimators for mortar finite element methods with Lagrange multipliers, SIAM J. Numer. Anal., 36 (1999) 1636-1658. MR 2000e:65104
  • 26. B. Wohlmuth, Discretization methods and iterative solvers based on domain decomposition, Habilitation Thesis, University of Augsburg, November 1999.

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Additional Information

W. Dahmen
Affiliation: Institut für Geometrie und Praktische Mathematik, RWTH Aachen, Templergraben 55, D-52062 Aachen, Germany

B. Faermann
Affiliation: Institut für Mathematik, Sekretariat MA 4-5, Technische Univerität Berlin, D-10623 Berlin, Germany

I. G. Graham
Affiliation: Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, United Kingdom

W. Hackbusch
Affiliation: Max-Planck-Institut Mathematik in den Naturwissenschaften, D-04103 Leipzig, Inselstr. 22-26, Germany

S. A. Sauter
Affiliation: Institut für Mathematik, Universität Zürich, Winterthurerstr 190, CH-8057 Zürich, Switzerland

Keywords: Inequalities, mesh-dependent norms, inverse estimates, nonlinear approximation theory, nonmatching grids, mortar element method, boundary element method
Received by editor(s): May 2, 2001
Received by editor(s) in revised form: January 10, 2003
Published electronically: July 14, 2003
Article copyright: © Copyright 2003 American Mathematical Society