External approximation of first order variational problems via W-1,p estimates

Cesare Davini; Roberto Paroni

ESAIM: Control, Optimisation and Calculus of Variations (2008)

  • Volume: 14, Issue: 4, page 802-824
  • ISSN: 1292-8119

Abstract

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Here we present an approximation method for a rather broad class of first order variational problems in spaces of piece-wise constant functions over triangulations of the base domain. The convergence of the method is based on an inequality involving W - 1 , p norms obtained by Nečas and on the general framework of Γ-convergence theory.

How to cite

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Davini, Cesare, and Paroni, Roberto. "External approximation of first order variational problems via W-1,p estimates." ESAIM: Control, Optimisation and Calculus of Variations 14.4 (2008): 802-824. <http://eudml.org/doc/250314>.

@article{Davini2008,
abstract = { Here we present an approximation method for a rather broad class of first order variational problems in spaces of piece-wise constant functions over triangulations of the base domain. The convergence of the method is based on an inequality involving $W^\{-1, p\}$ norms obtained by Nečas and on the general framework of Γ-convergence theory. },
author = {Davini, Cesare, Paroni, Roberto},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Numerical methods; non-conforming approximations; Γ-convergence; -convergence; estimates; discontinuous Galerkin method; Necas reverse inequality; convergence; discretized functionals},
language = {eng},
month = {1},
number = {4},
pages = {802-824},
publisher = {EDP Sciences},
title = {External approximation of first order variational problems via W-1,p estimates},
url = {http://eudml.org/doc/250314},
volume = {14},
year = {2008},
}

TY - JOUR
AU - Davini, Cesare
AU - Paroni, Roberto
TI - External approximation of first order variational problems via W-1,p estimates
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2008/1//
PB - EDP Sciences
VL - 14
IS - 4
SP - 802
EP - 824
AB - Here we present an approximation method for a rather broad class of first order variational problems in spaces of piece-wise constant functions over triangulations of the base domain. The convergence of the method is based on an inequality involving $W^{-1, p}$ norms obtained by Nečas and on the general framework of Γ-convergence theory.
LA - eng
KW - Numerical methods; non-conforming approximations; Γ-convergence; -convergence; estimates; discontinuous Galerkin method; Necas reverse inequality; convergence; discretized functionals
UR - http://eudml.org/doc/250314
ER -

References

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  1. B.A. Andreianov, M. Gutnic and P. Wittbold, Convergence of finite volume approximations for a nonlinear elliptic-parabolic problem: a “continuous" approach. SIAM J. Numer. Anal.42 (2004) 228–251.  Zbl1080.65081
  2. D.N. Arnold, An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal.19 (1982) 742–760.  Zbl0482.65060
  3. D.N. Arnold, F. Brezzi, B. Cockburn and L.D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal.39 (2001-2002) 1749–1779.  Zbl1008.65080
  4. J.P. Aubin, Approximation des problèmes aux limites non homogènes pour des opérateurs non linéaires. J. Math. Anal. Appl.30 (1970) 510–521.  Zbl0216.17103
  5. I. Babuška, The finite element method with penalty. Math. Comp.27 (1973) 221–228.  Zbl0299.65057
  6. I. Babuška and M. Zlámal, Nonconforming elements in the finite element method with penalty. SIAM J. Numer. Anal.10 (1973) 863–875.  Zbl0237.65066
  7. I. Babuška, C.E. Baumann and J.T. Oden, A discontinuous hp finite element method for diffusion problems: 1-D analysis. Comput. Math. Appl.37 (1999) 103–122.  Zbl0940.65076
  8. C.E. Baumann and J.T. Oden, Advances and applications of discontinuous Galerkin methods in CFD. Computational mechanics (Buenos Aires, 1998), Centro Internac. Métodos Numér. Ing., Barcelona (1998).  
  9. C.E. Baumann and J.T. Oden, A discontinuous hp finite element method for convection-diffusion problems. Comput. Methods Appl. Mech. Engrg.175 (1999) 311–341.  Zbl0924.76051
  10. C.E. Baumann and J.T. Oden, An adaptive-order discontinuous Galerkin method for the solution of the Euler equations of gas dynamics. Internat. J. Numer. Methods Engrg.47 (2000) 61–73.  Zbl0984.76040
  11. H. Brezis, Analyse fonctionnelle: Théorie et applications. Masson, Paris (1983).  Zbl0511.46001
  12. P.G. Ciarlet, The finite element method for elliptic problems. North Holland, Amsterdam (1978).  Zbl0383.65058
  13. P.G. Ciarlet, Basic error estimates for elliptic problems, in Handbook of numerical analysis, P.G. Ciarlet and J.-L. Lions Eds., North Holland, Amsterdam (1991).  Zbl0875.65086
  14. B. Cockburn and C.-W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal.35 (1998) 2440–2463.  Zbl0927.65118
  15. B. Cockburn, G.E. Karniadakis and C.-W. Shu, The development of discontinuous Galerkin methods, in Discontinuous Galerkin methods (Newport, RI, 1999), Lect. Notes Comput. Sci. Eng.11 (2000) 3–50.  Zbl0989.76045
  16. B. Dacorogna, Direct methods in the calculus of variations. Springer-Verlag, New York (1989).  Zbl0703.49001
  17. G. Dal Maso, An introduction to Γ-convergence. Birkäuser, Boston (1993).  
  18. C. Davini, Piece-wise constant approximations in the membrane problem. Meccanica38 (2003) 555–569.  Zbl1062.74561
  19. C. Davini and F. Jourdan, Approximations of degree zero in the Poisson problem. Comm. Pure Appl. Anal.4 (2005) 267–281.  Zbl1084.65114
  20. C. Davini and R. Paroni, Generalized Hessian and external approximations in variational problems of second order. J. Elasticity70 (2003) 149–174.  Zbl1049.65116
  21. C. Davini and R. Paroni, Error estimate of piece-wise constant approximations to the Poisson problem (in preparation).  Zbl1049.65116
  22. C. Davini and I. Pitacco, Relaxed notions of curvature and a lumped strain method for elastic plates. SIAM J. Numer. Anal.35 (1998) 677–691.  Zbl0928.74099
  23. C. Davini and I. Pitacco, An unconstrained mixed method for the biharmonic problem. SIAM J. Numer. Anal.38 (2000) 820–836.  Zbl0982.65124
  24. L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics. CRC Press, Boca Raton (1992).  Zbl0804.28001
  25. J.-L. Lions, Problèmes aux limites non homogènes à données irrégulières : Une méthode d'approximation, in Numerical Analysis of Partial Differential Equations (C.I.M.E. 2 Ciclo, Ispra, 1967), Edizioni Cremonese, Rome (1968) 283–292.  
  26. J. Ne c ˇ as, Équations aux dérivées partielles. Presses de l'Université de Montréal (1965).  
  27. J. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Sem. Univ. Hamburg36 (1971) 9–15.  Zbl0229.65079
  28. W.H. Reed and T.R. Hill, Triangular mesh method for neutron transport equation. Tech. Report LA-UR-73-479, Los Alamos Scientific Laboratory, Los Alamos (1973).  
  29. M.F. Wheeler, An elliptic collocation-finite element method with interior penalties. SIAM J. Numer. Anal.15 (1978) 152–161.  Zbl0384.65058
  30. X. Ye, A new discontinuous finite volume method for elliptic problems. SIAM J. Numer. Anal.42 (2004) 1062–1072.  Zbl1079.65116

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