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Nonconforming finite element approximation of crystalline microstructure

Authors: Bo Li and Mitchell Luskin
Journal: Math. Comp. 67 (1998), 917-946
MSC (1991): Primary 49M15, 65C20, 65N30, 73C50, 73K20
MathSciNet review: 1459391
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Abstract: We consider a class of nonconforming finite element approximations of a simply laminated microstructure which minimizes the nonconvex variational problem for the deformation of martensitic crystals which can undergo either an orthorhombic to monoclinic (double well) or a cubic to tetragonal (triple well) transformation. We first establish a series of error bounds in terms of elastic energies for the $L^2$ approximation of derivatives of the deformation in the direction tangential to parallel layers of the laminate, for the $L^2$ approximation of the deformation, for the weak approximation of the deformation gradient, for the approximation of volume fractions of deformation gradients, and for the approximation of nonlinear integrals of the deformation gradient. We then use these bounds to give corresponding convergence rates for quasi-optimal finite element approximations.

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Additional Information

Bo Li
Affiliation: Department of Mathematics University of California, Los Angeles 405 Hilgard Avenue Los Angeles, California 90095-1555

Mitchell Luskin
Affiliation: School of Mathematics University of Minnesota 206 Church Street, S.E. Minneapolis, Minnesota 55455

Keywords: Finite element, nonconforming, microstructure, martensitic transformation, error estimate
Received by editor(s): June 12, 1996
Received by editor(s) in revised form: February 26, 1997
Additional Notes: This work was supported in part by the NSF through grant DMS 95-05077, by the AFOSR through grants AF/F49620-96-1-0057 and AF/F49620-97-1-0187, by the Institute for Mathematics and its Applications, and by a grant from the Minnesota Supercomputer Institute.
Article copyright: © Copyright 1998 American Mathematical Society