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

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

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topLorent, 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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- [3] M. Chipot, The appearance of microstructures in problems with incompatible wells and their numerical approach. Numer. Math. 83 (1999) 325–352. Zbl0937.65070
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- [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). Zbl0819.28004MR1333890
- [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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