Analysis And Application Of The Characteristics-mixed Finite Element Method

This paper considers the numerical simulation of the convection-dominated diffusion problemand the integro-differential problemThe equation describing these problems is uniformly parabolic, but in many applications, the convection dominates diffusion, the equation is nearly hyperbolic in nature. It is well known that strictly parabolic discretization scheme applied to the problem do not work well, it tends to introduce into the solution an excessive amount of numerical diffusion near the fronts. To purse a high performance of an algorithm, we propose a new numerical scheme, called the characteristics-mixed finite element method, for approximating the solution to a convection-dominated diffusion equations of parabolic type. The new method is a combination of characteristic approximation to handle the convection part, to ensure the high stability of the method in approximating the sharp fronts and reduce the numerical diffusion, a smaller time truncation is gained at the same time, and a mixed finite elementspatial approximation to deal with the diffusion part, the sealer unknown and the adjoint vector function are approximated optimally and simultaneously.The method is firstly applied to solve the linear convection-dominated diffusion problems and the optimal L2-error estimates of the unknown function c and the vector flux p are gained. For the case of nonlinear diffusion part, we derived the same results as in the linear probloms case. Then we come to the nonlinear advection-dominated transport problem, only the one-dimensional case is discussed. It is shown that the unknown c and the adjoint p are optimally approximated when it is discretized with the lowest order mixed finite element space. Finally the convection-dominated integro-differential equations of parabolic type are considered, the optimal error estimates for the unknown c and the adjoint flux p gained in the previous parts are still hold. Numerical examples for the linear, quasi-linear and integro-differential problems show that the method we propose is practically effective.