| Interests have grown for the numerical solutions for a class of parabolic systems, because they were applied in many fields (such as: in biology, structural and fluid mechanics, and other fields ). It is a question cared by computable mathematics and applied mathematics that how effectively approximately to solve parabolic systems. Since the standard Galerkin finite element method firstly was applied to parabolic systems by J.Douglas,Jr. and T.Dupont in 1970, there have been much development in the field of numerical solution for the parabolic systems. Many methods had been builded. For example, The semi-discrete scheme, Crank-Nicolson scheme, finite element method with moving mesh and finite volume element method have been builded. In the reserach of numerical methods of oil reservoir numerical simulation, J.Dougals,Jr. introduced the characteristics in 1982. Since then, J.Douglas.Jr., R.E.Ewing, T.F.Russell, M.F.Wheeler and Yuan Yirang have completed a series of fundamental researchs in this field, and have presented many famous numerical methods, such as: the characteristics finite differernce method and the characteristics finite element method, and have completed theoretical analysis and numerical experiments that have become the basic theories. From the numerical experical experiments, we could find that the characteristics method could increase time step and avoid numerical oscillation, as opposed to general methods. In the past, the characteristics method was applied to solve the parabolic equation. Recently, there have been many advancements that it is applied to the parabolic systems. For example: Du Ning haved builded characteristics-economical difference scheme and alternating-direction characteristics finite elememt method for the parabolic systems.In this dissertation, for a class of parabolic systems in boundary domain, we present the characteristics difference method and the characteristics finite element method under the direction of professor Yuan Yirang, and give rigorous convergence analysis and experiments which verify the theoretical results and indicate the efficiency and validity of these schemes. In 2003, Du Ning presents characteristics-economical difference scheme for one-dimensional convection-diffusion coupled system. However, in my paper, I adaptquadratic interpolation instead of linear interpolation, and give the experiments, lhe characteristics finite element method is applied to parabolic systems in hypothesis of convection term coefficient. The method develops J.Douglas,Jr. work and enrichs research and application for the parabolic systems. The work consists of two chapters. The same model: Inl.3, we obtain two theorems :Theorem 1.1 (Stability theorem of the characteristics difference)If the parabolic systems satisfy the coefficient hypothesis(I). and At — O(h2), and Ui1(x) and U%l{x) are defined by quadratic interpolation . the solution of the characteristics difference scheme is stability.Theorem 1.2 (Convergence theorem of the characteristics difference)If the parabolic systems satisfy the coefficient hypothesis(I). and its solution is u e I°°(0, T; w4'2{Q)), and §￡ e L2(0, T; P) . U is the solution of the characteristics difference scheme, thenn -?■,|| 2 #■?II" -I1 \\ < k{\\u\\L0.T;w*i(0,l)h +\\^q:\\L2i0T.j2jAt)In 1.4, we give a numerical experiment.In chapter 2, we consider the characteristics finite element method for these systems. 2.1 is the same as the 1.1. In 2.2, we present the characteristics finite element scheme:,Vxv]dx = (f(x,tn+1),v) Vv€nh[U°, v) = (uo(x), v) Vu € HhIn 2.3, we obtain a theorem:Theorem 2.1 (Convergence theorem of the characteristics finite element method)We hypothesize u is the solution of the parabolic systems, and satisfy the coefficience hypothesis(II). U is the solution of the characteristics finite element schemes, thenMmax II(u - U)n\\2,2 + A*y ||(u - ￡/)n||ii < K{(At)2 + h2r+2)0 n=0In 2.4 , we give a numerical experiment.
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