A regularized–stabilized mixed finite element formulation for viscoplasticity of Bingham type

In this article, we present a new mixed stabilized–regularized finite element formulation in primitive variables, with continuous velocity and discontinuous pressure interpolations for the steady flow of an incompressible fluid of Bingham type. This formulation is based on an augmented Lagrangian regularization technique and a least squares stabilization method. Mathematical analyses are performed for the new formulation in terms of stability, existence and uniqueness of the solution. Optimal orders of convergence are obtained mathematically, improving on classical methods, which also present limitations as regards the values of the yield stress. Numerical results are presented confirming the theory developed here, and they show the robustness of the new method, with stability obtained for the velocity and, especially, for the pressure when the yield stress is very high.