A Uniform Optimal-Order Error Estimate Of An Expanded Characteristic-Mixed Finite Element Method For A Convection-Dominated Transport Problem

In this paper, vve propose a uniform optimal-order error estimate of an expand-ed mixed finite element method for a kind of elliptic diffusion problems with small diffusive parameter ε and prove a uniform optimal-order error estimate of an ex-panded characteristic-mixed finite element method for two-dimensional convection-dominated diffusion equation with small diffusive parameter ε.For the second order elliptic diffusion problems, in engineering practice, people not only care about pressure u, but also concern with σ=-K(x,y)▽u(x,y)-Darcy velocity. In order to approximate u and Darcy velocity σ at the same time, mixed finite element method and expanded mixed finite element method are proposed [1,7,8,9]. It is proved that these methods can approximate u and a with high pre-cision. Unfortunately the control constant C in the corresponding error estimates depends on the inverse of the small scaled parameter ε, which indicates that the mixed finite element solution will blow up as ε tends to zero.In order to overcome the defect of the above methods, In this paper we propose an expanded mixed finite element method and prove its solvability. Furthermore, we prove e-uniform optimal-order L2-norm error estimates for the proposed procedure, which means that the control constant C in the corresponding error estimates do not depend on the inverse of ε. It guarantees that our expanded mixed finite element method can successfully simulate diffusion process in low permeable area. In view of the superiority of this method, we apply it to the uniform optimal-order error estimate for two-dimensional convection-dominated diffusion equation with small diffusive parameter ε.For the convection-dominated diffusion equation, if the diffusive coefficient K is uniformly positive the equation is strictly parabolic. But in many practical ap-plications, K is a small enough, thus convection dominates diffusion, the problem is nearly hyperbolic in nature. There will be an excessive amount of numerical d-iffusion when approximate the sharp fronts. So the strictly parabolic discretization schemes applied to the problem do not work well when it is convection dominat-ed. In order to overcome the defects of the traditional methods, a series of high performance algorithms have been proposed by several authors, such as the explicit characteristic method, the streamline diffusion method[16,17], characteristic finite difference method and characteristic finite element method[5]. In order to better simulate such problems, people have developed the characteristic-mixed finite ele-ment method and the modified characteristics-mixed finite element method[4,18]. A great many numerical experiences show that these methods are practically ef-fective. They not only ensure the high stability of the method in approximating the sharp fronts and eliminate the numerical diffusion but also approximate the unknown function and the adjoint vector optimally and simultaneously. However, these error estimates were proved by introducing the mixed elliptic projection whose approximation property depends on the reciprocal of the small parameter ε, thus the error estimates derived in[4,18] also depend on ε-1. When ε small enough, the constants in the right of the estimator will be very large and lead to estimate failure.For deriving the optimal-order estimate in L2norm and make the estimates be independent of ε. In this paper, we adopt the expanded characteristic-mixed fi-nite element method for a convection-dominated transport problem with a periodic boundary condition. We improve the Lemmal and Lemma2[6] and use the inter-polation operator and Raviart-Thomas projection instead of the mixed elliptic projection, therefore the uniform error estimates for the unknown function, its flux and its gradient are achieved. The generic constants in the error estimates do not explicitly depend on e, but depend linearly on certain Sobolev norms of the true solution. Furthermore, combining the estimates with the stability estimates of the true solution, we prove these estimates depend only on the initial and the right side data. In the last Section, numerical experiments are performed to verify our theoretical findings.