Numerical approximations in variational problems with potential wells. (English) Zbl 0763.65049

The paper is concerned with numerical aspects of variational problems which fail to be convex. The infimum of the energy fails to be attained but minimizing sequences develop oscillations which allow them to decrease the energy. Different rates of convergence of the energy are given. The analysis of the oscillations of the minimizing sequences is made. In particular it is shown that the minimizing sequences choose their gradients in the vicinity of the wells with a probability which tends to be constant.
Such oscillations are observed in the context of hyperelasticity for ordered materials such as crystals. Numerical experiments are presented. The results of this paper extend the results obtained in one dimension by the second author, D. Kinderlehrer and M. Luskin [ibid. 28, No. 2, 321-332 (1991; Zbl 0725.65067)] and the second author and M. Luskin [Math. Comput. 57, No. 196, 621-637 (1991; Zbl 0735.65042)].
Reviewer: V.Arnautu (Iaşi)


65K10 Numerical optimization and variational techniques
49M15 Newton-type methods
49J40 Variational inequalities
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