A second-order linear scalar differential equation including a zeroth order term is approximated by using a new stablized finite element method, which can attenuate the oscillations present using the standard Galerkin method with very small coefficient of the seconder-order term. By the helpness of additional variable which could be static condensated, we generalize the "unusual stabilized finite element method" developed in [1] from linear triangle element to bilinear quartdral element, and we establish a convergence study and obtain successful numerical simulation.
This paper is concerned with the numerical analysis of viscous-elastic flow problems relating material creep deformation nature under high temperature. Early works in this area are those by Barret, Liu[1], and Baranger and Najib [2]. These paper studied those model consisting only velocity and pressure as variables.Recently, in view of the special importance of the stress in material property,Baranger studied new finite element formula for quasi-Newtonian flow obeying power law, which consists stress, velocity and pressure as variables. Assume the Newtonian part is zero (see [3]), and one LBB condition is satisfied between the stress and velocity finite element spaces. Baranger has proven that the discrete formula has unique solution, and the error estimation is also deduced. Considering the restrictive condition given by [3] for finite element spaces, we can learn that it is difficult to use this method in practise.In this paper we constructs 4 new three field model, which also regards stress,velocity and pressure as variables, for above quasi-Newtonian flow problem, we also prove the existence and uniqueness of finite element solution, and deduce ’related error estimation, only by assuming a LBB condition hold between the velocity and pressure finite element spaces. Compared with that one in [3], the restrictive condition between velocity and stress finite element spaces is removed,thus we apparently relaxed the restrictive condition demanded to finite element space combination satisfying all our assumptions is proposed.
