An Upwind Finite Volume Element Method for Nonlinear Convection Diffusion Problem

where is a bounded region with piecewise smooth boundary . R    is a small positive constant and is a smooth vector function on   u f   F x x  1 2 u        3 f u f u    x x  R  ,   0 0   F x . The finite volume element method (FVEM) is a discrete technique for partial differential equations, especially for those arising from physical conservation laws, including mass, momentum and energy. This method has been introduced and analyzed by R. Li and his collaborators since 1980s, see [1] for details. The FVEM uses a volume integral formulation of the original problem and a finite partitioning set of covolumes to discretize the equations.The approximate solution is chosen out of a finite element spaces [1-3] The FVEM is widely used in computational fluid mechanics and heat transfer problems [2-5]. It possesses the important and crucial property of inheriting the physical conservation laws of the original problem locally. Thus it can be expected to capture shocks, or to study other physical phenomena more effectively. On the other hand, the convection-dominated diffusion problem has strong hyperbolic characteristics, and therefore the numerical method is very difficult in mathematics and mechanics. when the central difference method, though it has second-order accuracy, is used to solve the convection-dominated diffusion problem, it produces numerical diffusion and oscillation near the discontinuous domain, making numerical simulation failure. The case usually occurs when the finite element methods (FEM) and FVEM are used for solve the convectiondominated diffusion problem. For the two-phase plane incompressible displacement problem which is assumed to be -periodic, J. Douglas, Jr., and T.F.Russell have published some articles on the characteristic finite difference method and FEM to solve the convection-dominated diffusion problems and to overcome oscillation and faults likely to occur in the traditional method [6]. Tabata and his collaborators have been studying upwind schemes based triangulation for convection-diffusion problem since 1977 [7-11]. Yuan, starting from the practical exploration and development of oil-gas resources, put forward the upwind finite difference fractional steps methods for the two-phase threedimensional compressible displacement problem [12]. 