An optimal scaling law for finite element approximations of a variational problem with non-trivial microstructure

Andrew Lorent

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2001)

  • Volume: 35, Issue: 5, page 921-934
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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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 35.5 (2001): 921-934. <http://eudml.org/doc/194081>.

@article{Lorent2001,
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 satisfy an affine boundary condition in the second lamination convex hull of the wells of the functional.},
author = {Lorent, Andrew},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {finite-well non-convex functionals; finite element approximations; optimal scaling law; variational problem; non-trivial microstructure; lower bounds; affine boundary condition; second lamination convex hull; triangular grid},
language = {eng},
number = {5},
pages = {921-934},
publisher = {EDP-Sciences},
title = {An optimal scaling law for finite element approximations of a variational problem with non-trivial microstructure},
url = {http://eudml.org/doc/194081},
volume = {35},
year = {2001},
}

TY - JOUR
AU - Lorent, Andrew
TI - An optimal scaling law for finite element approximations of a variational problem with non-trivial microstructure
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2001
PB - EDP-Sciences
VL - 35
IS - 5
SP - 921
EP - 934
AB - 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.
LA - eng
KW - finite-well non-convex functionals; finite element approximations; optimal scaling law; variational problem; non-trivial microstructure; lower bounds; affine boundary condition; second lamination convex hull; triangular grid
UR - http://eudml.org/doc/194081
ER -

References

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  1. [1] J.M. Ball and R.D. James, Fine phase mixtures as minimizers of energy. Arch. Rational Mech. Anal. 100 (1987) 13–52. Zbl0629.49020
  2. [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. Zbl0758.73009
  3. [3] M. Chipot, The appearance of microstructures in problems with incompatible wells and their numerical approach. Numer. Math. 83 (1999) 325–352. Zbl0937.65070
  4. [4] M. Chipot and D. Kinderlehrer, Equilibrium configurations of crystals. Arch. Rational Mech. Anal. 103 (1988) 237–277. Zbl0673.73012
  5. [5] G. Dolzmann, Personal communication. 
  6. [6] M. Luskin, On the computation of crystalline microstructure. Acta Numer. 5 (1996) 191–257. Zbl0867.65033
  7. [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. [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. [9] P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, in Cambridge Studies in Advanced Mathematics, Cambridge (1995). Zbl0819.28004MR1333890
  10. [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. Zbl0797.73079

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