Application of a first-order operator splitting method to Bingham fluid flow simulation

We discuss here a first-order operator splitting method for the solution of the time dependent variational inequality modeling the flow of a Bingham visco-plastic medium. At each time step, we shall have to solve a nonlinear elliptic problem, then a nondifferentiable minimization problem, and finally a Stokes type problem. The first problem is solved by a fixed point method; the second one happens to be a saddle point problem for a Lagrangian functional, and we shall use an Uzawa algorithm to solve it. The last problem is solved by a conjugate gradient method. We have applied this splitting method to two test problems: the first one has a forcing term with homogeneous Dirichlet boundary conditions, while the second one concerns the simulation of the flow of a Bingham fluid in a wall driven cavity. Our results compare very well with the ones obtained by solving directly the variational inequality mentioned above.