Approximation by Piecewise Constant Functions in a Bv Metric

In mathematical models for crystal microstructure [1,2,27], the deformation gradient is nearly piecewise constant in space to enable the deformation to attain a low energy. The length scale of the microstructure is limited by a surface energy associated with the transition from one piecewise constant variant phase to another piecewise constant variant phase [1, 27]. Motivated by these models, numerical methods have been developed and utilized that approximate the deformation gradient by piecewise constant functions and that minimize an energy that includes both an elastic energy and a surface energy proportional to the total variation of the deformation gradient [8]. In this paper, we give approximation results for piecewise constant functions in a metric for functions of bounded variation. We consider these results in the context of a thin film model for martensitic crystals. We have developed numerical methods for the computation of microstructure in martensitic and ferromagnetic crystals and validated these methods by the development of a numerical analysis of microstructure [4,6,7,14,27–30]. Related results are given in [10, 12, 14, 20–26,31, 32].