A Dual-Mixed Approximation for a Huber Regularization of the Herschel-Bulkey Flow Problem

In this paper, we extend a dual-mixed formulation for a nonlinear generalized Stokes problem to a Huber regularization of the Herschel-Bulkey flow problem. The present approach is based on a two-fold saddle point nonlinear operator equation for the corresponding weak formulation. We provide the uniqueness of solutions for the continuous formulation and propose a discrete scheme based on Arnold-Falk-Winther finite elements. The discretization scheme yields a system of Newton differentiable nonlinear equations, for which a semismooth Newton algorithm is proposed and implemented. Local superlinear convergence of the method is also proved. Finally, we perform several numerical experiments to investigate the behavior and efficiency of the method.