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Mathematics of Computation

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Criteria for the approximation property for multigrid methods in nonnested spaces

Authors: Nicolas Neuss and Christian Wieners
Journal: Math. Comp. 73 (2004), 1583-1600
MSC (2000): Primary 65N55, 65F10
Published electronically: March 9, 2004
MathSciNet review: 2059727
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Abstract | References | Similar Articles | Additional Information

Abstract: We extend the abstract frameworks for the multigrid analysis for nonconforming finite elements to the case where the assumptions of the second Strang lemma are violated. The consistency error is studied in detail for finite element discretizations on domains with curved boundaries. This is applied to prove the approximation property for conforming elements, stabilized $Q_1/P_0$-elements, and nonconforming elements for linear elasticity on nonpolygonal domains.

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  • 1. I. Babuska, C. Caloz, and J. Osborn, Special finite element methods for a class of second order elliptic problems with rough coefficients, SIAM J. Numer. Anal. 31 (1994), 945-981. MR 95g:65146
  • 2. F.-J. Barthold, M. Schmidt, and E. Stein, Error indicators and mesh refinements for finite-element-computations of elastoplastic deformations, Computational Mechanics 22 (1998), 225-238.
  • 3. P. Bastian, K. Birken, K. Johannsen, S. Lang, N. Neuß, H. Rentz-Reichert, and C. Wieners, UG - a flexible software toolbox for solving partial differential equations, Comp. Vis. Sci. 1 (1997), 27-40.
  • 4. D. Braess, M. Dryja, and W. Hackbusch, Multigrid method for nonconforming fe-discretisations with application to nonmatching grids, Computing 63 (1999), 1-25. MR 2000h:65048
  • 5. J. H. Bramble, Multigrid methods, Longman Scientific & Technical, Essex, 1993.MR 95b:65002
  • 6. J.H. Bramble and J. T. King, A robust finite element method for nonhomogeneous Dirichlet problems in domains with curved boundaries, Math. Comp. 63 (1994), 1-17. MR 94i:65112
  • 7. S. C. Brenner, Convergence of nonconforming multigrid methods without full elliptic regularity, Math. Comp. 68 (1999), 25-53. MR 99c:65229
  • 8. S. C. Brenner and R. Scott, The mathematical theory of finite element methods, Springer-Verlag, 1994. MR 95f:65001
  • 9. M. Crouzeix and V. Thomée, The stability in $L^p$ and $W^{1,p}$ of the $L^2$-projection onto finite element function spaces, Math. Comput. 48 (1987), 521-532. MR 88f:41016
  • 10. M. Lenoir, Optimal isoparametric finite elements and error estimates for domains involving curved boundaries, SIAM J. Numer. Anal. 23 (1986), 662-680. MR 87m:65163
  • 11. J. L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications, Springer-Verlag, 1972. MR 50:2670, MR 50:2671
  • 12. N. Neuß, Homogenisierung und Mehrgitterverfahren, Ph.D. thesis, Universität Heidelberg, 1995.
  • 13. J. C. Simo and T. J. R. Hughes, Computational inelasticity, Springer-Verlag, 1998. MR 99i:73038
  • 14. E. M. Stein, Singular integrals and differentiability properties of functions, Princeton mathematical series, vol. 30, Princeton Univ. Press, 1970.MR 44:7280
  • 15. R. Stevenson, An analysis of nonconforming multi-grid methods, leading to an improved method for the Morley element, Math. Comp. 72 (2003), 55-81.
  • 16. H. Triebel, Interpolation theory, function spaces, differential operators, Barth, 1995. MR 96f:46001
  • 17. C. Wieners, Multigrid methods for Prandtl-Reuß plasticity, Numer. Lin. Alg. Appl. 6 (1999), 457-478. MR 2000j:74092
  • 18. -, Robust multigrid methods for nearly incompressible elasticity, Computing 64 (2000), 289-306. MR 2001g:65168
  • 19. J. Wloka, Partielle Differentialgleichungen, Teubner, Stuttgart, 1982. MR 84a:35002
  • 20. M. Zlámal, Curved elements in the finite elements method I, SIAM J. Numer. Anal. 10 (1973), 229-240. MR 52:16060

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Additional Information

Nicolas Neuss
Affiliation: Universität Heidelberg, IWR, Im Neuenheimer Feld 368, 69120 Heidelberg, Germany

Christian Wieners
Affiliation: Universität Karlsruhe (TH), Institut für Praktische Mathematik, Engesser Str. 2, 76128 Karlsruhe, Germany

Keywords: Multigrid analysis, nonnested forms, approximation property, curved boundaries, stabilized finite elements
Received by editor(s): January 23, 2001
Received by editor(s) in revised form: March 21, 2003
Published electronically: March 9, 2004
Additional Notes: This work was supported in part by the Deutsche Forschungsgemeinschaft
Article copyright: © Copyright 2004 American Mathematical Society

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