Least-Squares Finite Element Methods For Non-Newtonian Viscoelastic Fluid Models

Non-Newtonian viscoelastic flows are found in several biological, material and industrial applications, such as polymer processes, coating and extrusion of polymeric material and artificial organs. Due to the hyperbolic character of the constitutive equation, numerical simulation of vicoelastic flows is a difficult task. Thus it is very important to establish the stable and effective algorithms. In this paper, we propose the least-squares finite element methods for the PTT and UCM viscoelastic models.Firstly, the decoupled algorithm for solving PTT viscoelastic fluid by finite element method is presented. The differential model is decoupled into two small problems. The method consists to solve alternatively a Stokes-like problem by the weighted least-squares(WLS) finite element method, and the constitutive equation by the SUPG method. A priori error estimate for the WLS/SUPG finite element method is derived. The existence and uniqueness of the approximate solution are obtained by using fixed point techniques. We show that the mapping of the decoupled algorithm is locally contracting. The planar channel flow problem is used to illustrate our theoretical results.Secondly, the weighted least-squares finite element method for the viscoelastic fluid obeying PTT model is introduced. Because the constitutive equation is nonlinear, we use the approximations of the unknowns to linearize the nonlinear terms and obtain the linearized PTT model. The WLS finite element method is applied to solve the linear model. The least-squares functional contains the 2 norm residuals of each equation multiplied by a proper weight. The error estimates for the finite element solution are discussed. The numerical results show that this algorithm is convergent.At last, we propose a nonlinear weighted least-squares finite element method to solve the UCM viscoelastic fluid. By linearizing the constitutive equation and the momentum equation, we obtain the linear UCM model. A series of the finite element solutions to the linear model are used to approximate the exact solution of the nonlinear model. The continuity and coercivity of the homogeneous functional are shown. Then we show the existence of the nonlinear weighted least-squares finite element solution and analyze its error estimates.