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50, 153– 171 (2005) [9] R. Henrion, A. Jourani, J. Outrata, On the calmness of a class of multifunctions. SIAM J. Opt. 13, 603–618 (2002) [10] T. Kozubek, A. Markopoulos, T. Brzobohatý, R. Kuˇcera, V. Vondrák, Z. Dostál, MatSol MATLAB efficient solvers for problems in engineering. S. Mordukhovich, Generalized differential calculus for nonsmooth and set-valued mappings. J. Math. Anal. Appl. S. Mordukhovich, Variational Analysis and Generalized Differentiation, I: Basic Theory, II: Applications. Grundlehren Series (Fundamental Principles of Mathematical Sciences), Vols.

In Sect. 5 we discretise the iterative method and prove another convergence result. We also look at a finite element discretisation for a particular choice of S . In Sect. 6 we demonstrate that implementations of the iterative method work well in 1 and 2 dimensions. In Sect. 7 the performance of using both potentials is compared in detail for a 1D problem. In Appendix A we describe how we choose the parameters in our model for the numerics. 7) with the smooth double well and obstacle potentials, it is advantageous to introduce an abstract framework that both problems fit into.

It is discussed in [33] and [34]. Approximation of Mumford-Shah. Chambolle and Del Maso [12] and related papers prove €-convergence results for finite element approximations of the Mumford-Shah functional. These results have some relation to the convergence results that we obtain using a different approach. Our work differs from existing work, and hence offers a new contribution, in the following respects: • We introduce the phase field approximation to the model right from the start (rather than at the last minute in order to allow numerical simulations).

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