Abstract. The paper examines the one-dimensional total variation flow equation with Dirichlet boundary conditions. Thanks to a new concept of "almost classical" solutions we are able to determine evolution of facets -flat regions of solutions. A key element of our approach is the natural regularity determined by the nonlinear elliptic operator, for which x 2 is an example of an irregular function. Such a point of view allows us to construct solutions. We apply this idea to numerical simulations for typical initial data. Due to the nature of Dirichlet data, any monotone function is an equilibrium. We prove that each solution reaches such a steady state in finite time.

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