Least-squares finite element methods for generalized Newtonian and viscoelastic flows

The goal of this work is to develop a least-squares finite element approach for the equations governing generalized Newtonian and viscoelastic flows such as those occurring in polymer processes. The Carreau generalized Newtonian model and the Giesekus viscoelastic constitutive equation are considered. The least-squares method offers the advantages of always generating a symmetric positive definite system, insensitivity to equation type, and no need for compatibility conditions between finite element spaces. Results of the weighted least-squares approach are presented, along with comparisons using a Galerkin mixed method. The numerical results indicate that with carefully chosen nonlinear weighting functions, the least-squares solution achieves an optimal convergence rate in the L2-norm for the approximation to each dependent variable for the generalized Newtonian problem. For the viscoelastic flow problem, the numerical solution exhibits a second order convergence rate for the velocity and superlinear convergence rate in stress and pressure. Observed convergence rates depend strongly on the choice of weighting functions. Least-squares solutions for flows through a 4-to-1 contraction channel are also considered.