Convergence of conforming approximations for inviscid incompressible Bingham fluid flows and related problems

Abstract

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 nonsmooth 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.