A combined BDF-semismooth Newton approach for time-dependent Bingham flow

Abstract

This paper is devoted to the numerical simulation of time-dependent convective Bingham flow in cavities. Motivated by a primal-dual regularization of the stationary model, a family of regularized time-dependent problems is introduced. Well posedness of the regularized problems is proved and convergence of the regularized solutions to a solution of the original multiplier system is verified. For the numerical solution of each regularized multiplier system, a fully-discrete approach is studied. A stable finite element approximation in space, together with a second order backward differentiation formula for the time discretization are proposed. The discretization scheme yields a system of Newton differentiable nonlinear equations in each time step, for which a semismooth Newton algorithm is utilized. We present two numerical experiments to verify the main properties of the proposed approach.