A stable and scalable hybrid solver for rate-type non-Newtonian fluid models

We present and analyze hybrid discretization schemes for rate-type non-Newtonian fluids models. The method employs higher order conforming approximations for velocity and pressure of the Stokes equation and lower order approximations such as piecewise linear or piecewise constant for conformation tensor. To reduce the accuracy gap, the constitutive equation is discretized on a refined mesh, which is obtained by subdividing the mesh for the velocity and pressure. The temporal discretization is made by the standard semi-Lagrangian scheme. The fully discrete nonlinear system is shown to be solved iteratively by applying the three steps:(1) locating the characteristic feet of fluid particles, (2) solving the constitutive equation, and (3) solving the momentum and continuity equations (Stokes-type equation). To achieve a scalability of the solution process, we employ an auxiliary space preconditioning method for the solution to the conforming finite element methods for the Stokes equation. This method is basically two-grid method, in which lower-order finite element spaces are employed as auxiliary spaces. It is shown to not only lead to the mesh independent convergence, but also improve robustness and scalability. Stability analysis shows that if Δt=O(hd)Δt=O(hd), where dd is the dimension of domain, then the scheme admits a globally unique solution. A number of full 3D test cases are provided to demonstrate the advantages of the proposed numerical techniques in relation to efficiency, robustness, and weak scalability.