Parallel domain decomposition method for finite element approximation of 3D steady state non-Newtonian fluids

We introduce a stabilized finite element method for the 3D non-Newtonian Navier-Stokes equations and a parallel domain decomposition method for solving the sparse system of nonlinear equations arising from the discretization. Non-Newtonian flow problems are, generally speaking, more challenging than Newtonian flows because the nonlinearities are not only in the convection term but also in the viscosity term, which depends on the shear rate. Many good iterative methods and preconditioning techniques that work well for the Newtonian flows do not work well for the non-Newtonian flows. We employ a Galerkin/least squares finite element method, with stabilization parameters adjusted to count the non-Newtonian effect, to discretize the equations, and the resulting highly nonlinear system of equations is solved by a Newton-Krylov-Schwarz algorithm. In this study, we apply the proposed method to some inelastic power-law fluid flows through the eccentric annuli with inner cylinder rotation and investigate the robustness of the method with respect to some physical parameters, including the power-law index and the Reynolds number ratios. We then report the superlinear speedup achieved by the domain decomposition algorithm on a computer with up to 512 processors.