Hostname: page-component-7bb8b95d7b-fmk2r Total loading time: 0 Render date: 2024-09-07T08:16:13.775Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  30 January 2008

Cesare Davini  and
Roberto Paroni
Cesare Davini
Affiliation:
Dipartimento di Georisorse e Territorio, Via del Cotonificio 114, 33100 Udine, Italy; davini@uniud.it
Roberto Paroni
Affiliation:
Dipartimento di Architettura e Pianificazione, Università di Sassari, Palazzo del Pou Salit, Piazza Duomo, 07041 Alghero, Italy; paroni@uniss.it
Get access

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.

Keywords

Numerical methodsnon-conforming approximationsΓ-convergence
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 14 , Issue 4 , October 2008 , pp. 802 - 824
Copyright
© EDP Sciences, SMAI, 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Andreianov, B.A., Gutnic, M. and Wittbold, P., Convergence of finite volume approximations for a nonlinear elliptic-parabolic problem: a “continuous" approach. SIAM J. Numer. Anal. 42 (2004) 228–251. CrossRef
Arnold, D.N., An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19 (1982) 742–760. CrossRef
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.
Aubin, J.P., Approximation des problèmes aux limites non homogènes pour des opérateurs non linéaires. J. Math. Anal. Appl. 30 (1970) 510–521. CrossRef
Babuška, I., The finite element method with penalty. Math. Comp. 27 (1973) 221–228. CrossRef
Babuška, I. and Zlámal, M., Nonconforming elements in the finite element method with penalty. SIAM J. Numer. Anal. 10 (1973) 863–875. CrossRef
Babuška, I., Baumann, C.E. and Oden, J.T., A discontinuous hp finite element method for diffusion problems: 1-D analysis. Comput. Math. Appl. 37 (1999) 103–122. CrossRef
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).
Baumann, C.E. and Oden, J.T., A discontinuous hp finite element method for convection-diffusion problems. Comput. Methods Appl. Mech. Engrg. 175 (1999) 311–341. CrossRef
Baumann, C.E. and An, J.T. Oden adaptive-order discontinuous Galerkin method for the solution of the Euler equations of gas dynamics. Internat. J. Numer. Methods Engrg. 47 (2000) 61–73. 3.0.CO;2-D>CrossRef
H. Brezis, Analyse fonctionnelle: Théorie et applications. Masson, Paris (1983).
P.G. Ciarlet, The finite element method for elliptic problems. North Holland, Amsterdam (1978).
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).
Cockburn, B. and Shu, C.-W., The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35 (1998) 2440–2463. CrossRef
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. CrossRef
B. Dacorogna, Direct methods in the calculus of variations. Springer-Verlag, New York (1989).
G. Dal Maso, An introduction to Γ-convergence. Birkäuser, Boston (1993).
Davini, C., Piece-wise constant approximations in the membrane problem. Meccanica 38 (2003) 555–569. CrossRef
Davini, C. and Jourdan, F., Approximations of degree zero in the Poisson problem. Comm. Pure Appl. Anal. 4 (2005) 267–281.
Davini, C. and Paroni, R., Generalized Hessian and external approximations in variational problems of second order. J. Elasticity 70 (2003) 149–174. CrossRef
C. Davini and R. Paroni, Error estimate of piece-wise constant approximations to the Poisson problem (in preparation).
Davini, C. and Pitacco, I., Relaxed notions of curvature and a lumped strain method for elastic plates. SIAM J. Numer. Anal. 35 (1998) 677–691. CrossRef
Davini, C. and Pitacco, I., An unconstrained mixed method for the biharmonic problem. SIAM J. Numer. Anal. 38 (2000) 820–836. CrossRef
L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics. CRC Press, Boca Raton (1992).
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.
J. Ne $\check{\mbox{c}}$ as, Équations aux dérivées partielles. Presses de l'Université de Montréal (1965).
Nitsche, J., Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Sem. Univ. Hamburg 36 (1971) 9–15. CrossRef
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).
Wheeler, M.F., An elliptic collocation-finite element method with interior penalties. SIAM J. Numer. Anal. 15 (1978) 152–161. CrossRef
Ye, X., A new discontinuous finite volume method for elliptic problems. SIAM J. Numer. Anal. 42 (2004) 1062–1072. CrossRef