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Optimal-order error estimates for the finite element approximation of the solution of a nonconvex variational problem. (English) Zbl 0735.65042
We are dealing with a finite element method for some nonconvex variational problems arising in the equilibrium of crystal or other ordered materials. The bulk energy of a martensitic one-dimensional crystal is defined by the integral of a function of the scalar-valued displacement and of its derivative; and the main purpose of the paper is to give an optimal-order error analysis for the minimization of this energy over a finite element space. The Dirichlet problem is considered. Some results on the convergence of the deformation gradient are stated. Comparison with previous results in the literature is done.
Reviewer: G.Jumarie

MSC:
65K10 Numerical optimization and variational techniques
49M15 Newton-type methods
49K20 Optimality conditions for problems involving partial differential equations
35J20 Variational methods for second-order elliptic equations
35J70 Degenerate elliptic equations
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