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Nonconforming finite element approximation of crystalline microstructure. (English) Zbl 0901.73076
Summary: 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.

##### MSC:
 74S05 Finite element methods applied to problems in solid mechanics 74A60 Micromechanical theories 74M25 Micromechanics of solids 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 74A15 Thermodynamics in solid mechanics
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