A nonlinear weighted least-squares finite element method for the Oldroyd-B viscoelastic flow

This paper studies a nonlinear weighted least-squares finite element method for the steady Oldroyd-B viscoelastic flow. Our least-squares functional involves the L2-norm of the residuals of each equation multiplied by a proper weight. The weights include mass conservation constant, mesh dependent weight, and a nonlinear weighting function. We prove that the least-squares approximation converges to the solutions of the linearized versions of the viscoelastic fluid model at the best possible rate and then present the planar channel flow problem illustrating our theoretical results. Results of the least-squares approach indicate that with carefully chosen nonlinear weighting functions and linear basis functions, the numerical solution exhibits a second order convergence rate for velocity and viscous stress and superlinear convergence rate in polymeric stress and pressure. The method is applied to the 4-to-1 planar contraction problem. The effects of Weissenberg numbers are also presented in the work.