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

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- 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 35.5 (2010): 921-934. <http://eudml.org/doc/197548>.

@article{Lorent2010,

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},

keywords = {Finite-well non-convex functionals; finite element approximations.; 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},

month = {3},

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/197548},

volume = {35},

year = {2010},

}

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

DA - 2010/3//

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.; 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/197548

ER -

## References

top- J.M. Ball and R.D. James, Fine phase mixtures as minimizers of energy. Arch. Rational Mech. Anal.100 (1987) 13-52. Zbl0629.49020
- 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. A338 (1992) 389-450. Zbl0758.73009
- M. Chipot, The appearance of microstructures in problems with incompatible wells and their numerical approach. Numer. Math.83 (1999) 325-352. Zbl0937.65070
- M. Chipot and D. Kinderlehrer, Equilibrium configurations of crystals. Arch. Rational Mech. Anal.103 (1988) 237-277. Zbl0673.73012
- G. Dolzmann, Personal communication.
- M. Luskin, On the computation of crystalline microstructure. Acta Numer.5 (1996) 191-257. Zbl0867.65033
- 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). Zbl0819.28004
- 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. Zbl0797.73079

## Citations in EuDML Documents

top- Andrew Lorent, A Two Well Liouville Theorem
- Andrew Lorent, A two well Liouville theorem
- Andrew Lorent, The regularisation of the -well problem by finite elements and by singular perturbation are scaling equivalent in two dimensions
- Andrew Lorent, The regularisation of the $N$-well problem by finite elements and by singular perturbation are scaling equivalent in two dimensions

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