Finite element approximation of singular power-law systems

Non-Newtonian fluid motions are often modeled by a power-law ansatz. In the present paper, we consider shear thinning singular power-law models which feature an unbounded viscosity in the limit of zero shear rate, and we study the finite element (FE) discretization of the equations of motion. In the case under consideration, numerical instabilities usually arise when the FE equations are solved via Newton's method. In this paper, we propose a numerical method that enables the stable approximation of singular power-law systems and that is based on a simple regularization of the power-law model. Our proposed method generates a sequence of discrete functions that is computable in practice via Newton's method and that converges to the exact solution of the power-law system for diminishing mesh size. First, for the regularized model we discuss Newton's method and we show its stability in the sense that we derive an upper bound for the condition number of the Newton matrix. Then, we prove a priori error estimates that quantify the convergence of the proposed method. Finally, we illustrate numerically that our regularized approximation method surpasses the nonregularized one regarding accuracy and numerical efficiency.