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Optimal-order error estimates for the finite element approximation of the solution of a nonconvex variational problem


Authors: Charles Collins and Mitchell Luskin
Journal: Math. Comp. 57 (1991), 621-637
MSC: Primary 65N15; Secondary 35J20, 35J70, 65N30, 73C99, 73V05
DOI: https://doi.org/10.1090/S0025-5718-1991-1094944-0
MathSciNet review: 1094944
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Abstract: Nonconvex variational problems arise in models for the equilibria of crystals and other ordered materials. The solution of these variational problems must be described in terms of a microstructure rather than in terms of a deformation. Moreover, the numerical approximation of the deformation gradient often does not converge strongly as the mesh is refined. Nevertheless, the probability distribution of the deformation gradients near each material point does converge. Recently we introduced a metric to analyze this convergence. In this paper, we give an optimal-order error estimate for the convergence of the deformation gradient in a norm which is stronger than the metric used earlier.


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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1991-1094944-0
Keywords: Finite element method, error estimates, numerical approximation, variational problem, nonconvex
Article copyright: © Copyright 1991 American Mathematical Society