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

top
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

top

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

top
  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.  
  2. D.N. Arnold, An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal.19 (1982) 742–760.  
  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.  
  4. J.P. Aubin, Approximation des problèmes aux limites non homogènes pour des opérateurs non linéaires. J. Math. Anal. Appl.30 (1970) 510–521.  
  5. I. Babuška, The finite element method with penalty. Math. Comp.27 (1973) 221–228.  
  6. I. Babuška and M. Zlámal, Nonconforming elements in the finite element method with penalty. SIAM J. Numer. Anal.10 (1973) 863–875.  
  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.  
  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.  
  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.  
  11. H. Brezis, Analyse fonctionnelle: Théorie et applications. Masson, Paris (1983).  
  12. P.G. Ciarlet, The finite element method for elliptic problems. North Holland, Amsterdam (1978).  
  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).  
  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.  
  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.  
  16. B. Dacorogna, Direct methods in the calculus of variations. Springer-Verlag, New York (1989).  
  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.  
  19. C. Davini and F. Jourdan, Approximations of degree zero in the Poisson problem. Comm. Pure Appl. Anal.4 (2005) 267–281.  
  20. C. Davini and R. Paroni, Generalized Hessian and external approximations in variational problems of second order. J. Elasticity70 (2003) 149–174.  
  21. C. Davini and R. Paroni, Error estimate of piece-wise constant approximations to the Poisson problem (in preparation).  
  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.  
  23. C. Davini and I. Pitacco, An unconstrained mixed method for the biharmonic problem. SIAM J. Numer. Anal.38 (2000) 820–836.  
  24. L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics. CRC Press, Boca Raton (1992).  
  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.  
  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.  
  30. X. Ye, A new discontinuous finite volume method for elliptic problems. SIAM J. Numer. Anal.42 (2004) 1062–1072.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.