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

We study approximations by conforming methods of the solution to the variational inequality 〈∂tu, v − u〉 + ψ(v) − ψ(u) ≥ 〈f, v − u〉, which arises in the context of inviscid incompressible Bingham fluid flows and of the total variation flow problem. We propose a general framework involving total variation functionals, that enables to prove convergence of space, or time-space approximations, for steady or transient problems. We consider time implicit, or time implicit regularized (linearized or not) algorithms, and prove their convergence for general total variation functionals. Comparison with analytical solutions show the accuracy of the methods.