Nonconforming finite element approximation of crystalline microstructure

Bo Li, Mitchell Luskin

Research output: Contribution to journalArticlepeer-review

28 Scopus citations


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 L2 approximation of derivatives of the deformation in the direction tangential to parallel layers of the laminate, for the L2 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.

Original languageEnglish (US)
Pages (from-to)917-946
Number of pages30
JournalMathematics of Computation
Issue number223
StatePublished - Jul 1998


  • Error estimate
  • Finite element
  • Martensitic transformation
  • Microstructure
  • Nonconforming

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