Strong polyhedral approximation of simple jump sets

We prove a strong approximation result for functions u∈W1,∞(Ω∖J), where J is the union of finitely many Lipschitz graphs satisfying some further technical assumptions. We approximate J by a polyhedral set in such a manner that a regularisation term η(Divjui), (i=0,1,2,…), is convergent. The boundedness of this regularisation functional itself, introduced in [T. Valkonen, Transport equation and image interpolation with SBD velocity fields, J. Math. Pures Appl. 95 (2011) 459-494. doi:10.1016/j.matpur.2010.10.010 ensures the convergence in total variation of the jump part Divjui of the distributional divergence.