We study approximations by conforming methods of the solution to the variational inequality $\langle \partial_t u,v-u\rangle + \psi(v) - \psi(u) \ge \langle f,v-u\rangle$, which arises in the context of inviscid incompressible Bingham type fluid flows and of the total variation flow problem.
In the general context of a convex lower semi-continuous functional $\psi$ on a Hilbert space, we prove the convergence of time implicit space conforming approximations, without viscosity and for non-smooth data.
Then we introduce a general class of total variation functionals $\psi$, for which we can apply
the regularization method. We consider the time implicit regularized, linearized or not, algorithms,
and prove their convergence for general total variation functionals.
A comparison with an analytical solution concludes this study.

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