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

Andrew Lorent

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • 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 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

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  1. J.M. Ball and R.D. James, Fine phase mixtures as minimizers of energy. Arch. Rational Mech. Anal.100 (1987) 13-52.  
  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. A338 (1992) 389-450.  
  3. M. Chipot, The appearance of microstructures in problems with incompatible wells and their numerical approach. Numer. Math.83 (1999) 325-352.  
  4. M. Chipot and D. Kinderlehrer, Equilibrium configurations of crystals. Arch. Rational Mech. Anal.103 (1988) 237-277.  
  5. G. Dolzmann, Personal communication.  
  6. M. Luskin, On the computation of crystalline microstructure. Acta Numer.5 (1996) 191-257.  
  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:  
  9. P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, in Cambridge Studies in Advanced Mathematics, Cambridge (1995).  
  10. 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.  

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