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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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Nonconforming finite element approximation of crystalline microstructure
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by Bo Li and Mitchell Luskin PDF
Math. Comp. 67 (1998), 917-946 Request permission

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
  • Email: bli@math.ucla.edu
  • Mitchell Luskin
  • Affiliation: School of Mathematics University of Minnesota 206 Church Street, S.E. Minneapolis, Minnesota 55455
  • Email: luskin@math.umn.edu
  • 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.
  • © Copyright 1998 American Mathematical Society
  • Journal: Math. Comp. 67 (1998), 917-946
  • MSC (1991): Primary 49M15, 65C20, 65N30, 73C50, 73K20
  • DOI: https://doi.org/10.1090/S0025-5718-98-00941-7
  • MathSciNet review: 1459391