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Mathematical proofs are presented for the derivative superconvergence obtained by a class of patch recovery techniques for both linear and bilinear finite elements in the approximation of second-order elliptic problems.

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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.