Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-24T03:48:51.742Z Has data issue: false hasContentIssue false
  • English
  • Français

An optimal scaling law for finite element approximationsof a variationalproblem with non-trivial microstructure

Published online by Cambridge University Press:  15 April 2002

Andrew Lorent
Andrew Lorent*
Affiliation:
Max-Planck-Institute, Inselstr. 22-26, 04103 Leipzig, Germany. ([email protected])
Get access

Abstract

In this note we give sharp lower bounds for a non-convex functional when minimised over the space of functions that are piecewise affine on a triangular grid and satisfyan affine boundary condition in the second lamination convex hull of the wells of the functional.

Keywords

Finite-well non-convex functionalsfinite element approximations.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 35 , Issue 5 , September 2001 , pp. 921 - 934
Copyright
© EDP Sciences, SMAI, 2001

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

Ball, J.M. and James, R.D., Fine phase mixtures as minimizers of energy. Arch. Rational Mech. Anal. 100 (1987) 13-52. CrossRef
Ball, J.M. and James, R.D., Proposed experimental tests of a theory of fine microstructure and the two well problem. Philos. Trans. Roy. Soc. London Ser. A 338 (1992) 389-450. CrossRef
Chipot, M., The appearance of microstructures in problems with incompatible wells and their numerical approach. Numer. Math. 83 (1999) 325-352. CrossRef
Chipot, M. and Kinderlehrer, D., Equilibrium configurations of crystals. Arch. Rational Mech. Anal. 103 (1988) 237-277. CrossRef
G. Dolzmann, Personal communication.
Luskin, M., On the computation of crystalline microstructure. Acta Numer. 5 (1996) 191-257. CrossRef
M. Chipot and S. Müller, Sharp energy estimates for finite element approximations of non-convex problems. Variations of domain and free-boundary problems in solid mechanics, in Solid Mech. Appl. 66 , P. Argoul, M. Fremond and Q.S. Nguyen, Eds., Paris (1997) 317-325; Kluwer Acad. Publ., Dordrecht (1999).
Variational models for microstructure and phase transitions. MPI Lecture Note 2 (1998). Also available at:
P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, in Cambridge Studies in Advanced Mathematics, Cambridge (1995).
V. Sverák, On the problem of two wells. Microstructure and phase transitions. IMA J. Appl. Math. 54 , D. Kinderlehrer, R.D. James, M. Luskin and J. Ericksen, Eds., Springer, Berlin (1993) 183-189.