The Model Of Viscoelastic Fluid Flow And Its Numerical Analysis

Lately,the viscoelastic fluid flow is one of the most important questions in Hydrodynamics and theoretic mathematics.In this paper,we study one kind of viscoelastic fluid flow model.It is Oldroyd-B type fluid,the models can express using PDEs in math,and their solutions have received a great deal of attention.So,it is necessary to study highly efficient and highly accurate algorithms for PDEs,we present some kinds of method for solving PDEs using computational symbolic manipulation and finite element method.This paper includes five parts.In chapter 1,we introduce the theoretic basis of non-Newtonian fluid mechanics and rheology,and we introduce the mathematical foundation of Finite Element Method.In chapter 2,we set up the constitutive equation of Oldroyd-B type fluid based on Oldroyd self-time differential,present the grade 1,grade 2 variational analytical solutions respectively,specially,we present the concrete variational analytical solutions under constant pressure grade and periodic pressure grade.In chapter 3,our essential work is to solve viscoelastic fluid flow obeying an Oldroyd B type constitutive law by apply the mixed finite element method, least-square mixed finite element method and V-cycle multi-grid method.We study the mixed finite element method of Oldroyd B type viscoelastic fluid flow model,and we give the existence and uniqueness of approximation solution and error bounded; by applying the least-square mixed finite element methodto study Oldroyd B type viscoelastic fluid flow model,we discuss the existence and uniqueness of approximation solution and convergence;we analysis the V-cycle multi-grid formulation of Oldroyd B type viscoelastic fluid flow model,and we give the existence and uniqueness of iterative solution and error estimates.In chapter 4,by using least-squares mixed finite element method over quadrilaterals, we investigate super convergence phenomena sep-arately for boundary-value problems of un-symmetric elliptic equations,we obtain the super convergence result of least-squares mixed finite element solu-tions on the basis of the pr L~2 -projection and some mixed finite element projections and the integral identities technique developed by Q.Lin and his collaborates.In chapter 5,we present some two-grid methods for solving two-dimensional reaction-diffusion using expanded mixed finite element method,and we make our efforts to prove the convergence of the algorithms. We know the algorithms achieve asymptotically optimal approximation applying the two-grid methods.