We deal with a nonlinear hyperbolic scalar conservation law, regularized by the total variation flow
operator (or 1-Laplacian). We give an entropy weak formulation, for which we prove the existence and
the uniqueness of the solution. The existence result is established using the convergence of a numerical
approximation (a splitting scheme where the hyperbolic flow is treated with finite volumes and the total
variation flow with finite elements). Some numerical simulations are also presented.
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