An optimal scaling law for finite element approximations of a variational problem with non-trivial microstructure
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 35 (2001) no. 5, p. 921-934

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 satisfy an affine boundary condition in the second lamination convex hull of the wells of the functional.

Classification:  74B20,  74S05
Keywords: finite-well non-convex functionals, finite element approximations
@article{M2AN_2001__35_5_921_0,
author = {Lorent, Andrew},
title = {An optimal scaling law for finite element approximations of a variational problem with non-trivial microstructure},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {35},
number = {5},
year = {2001},
pages = {921-934},
zbl = {1017.74067},
mrnumber = {1866275},
language = {en},
url = {http://www.numdam.org/item/M2AN_2001__35_5_921_0}
}

Lorent, Andrew. An optimal scaling law for finite element approximations of a variational problem with non-trivial microstructure. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 35 (2001) no. 5, pp. 921-934. http://www.numdam.org/item/M2AN_2001__35_5_921_0/

[1] J.M. Ball and R.D. James, Fine phase mixtures as minimizers of energy. Arch. Rational Mech. Anal. 100 (1987) 13-52. | Zbl 0629.49020

[2] J.M. Ball and R.D. James, 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. | Zbl 0758.73009

[3] M. Chipot, The appearance of microstructures in problems with incompatible wells and their numerical approach. Numer. Math. 83 (1999) 325-352. | Zbl 0937.65070

[4] M. Chipot and D. Kinderlehrer, Equilibrium configurations of crystals. Arch. Rational Mech. Anal. 103 (1988) 237-277. | Zbl 0673.73012

[5] G. Dolzmann, Personal communication.

[6] M. Luskin, On the computation of crystalline microstructure. Acta Numer. 5 (1996) 191-257. | Zbl 0867.65033

[7] 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).

[8] Variational models for microstructure and phase transitions. MPI Lecture Note 2 (1998). Also available at: www.mis.mpg.de/cgi-bin/lecturenotes.pl

[9] P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, in Cambridge Studies in Advanced Mathematics, Cambridge (1995). | MR 1333890 | Zbl 0819.28004

[10] V. Šverák, 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. | Zbl 0797.73079