Two Types Of Evolution Equations, Some Numerical Methods

In this pater, we define a new numerical method called the characteristic mixed finite element method for approximating the solution to two kinds of evolution equation.In chapter one, we propose a new mixed method called characteristics mixed finite element method for a convection-dominated diffusion problems with small parameter e:We handle the convection part whth backward difference scheme along the characteristics, obtain much smaller time-trunction errors and avoid numerical dispersion on the front of the peak curve of the flow: we use a lowest order mixed finite element method to deal with the diffusion part, so this scheme can approximate the unknow function and its following vector with high accuracy at the same time. Piecewise constants are then in the test function space, so mass is cinserved element by element in the discrete level. The optimal L2 error estimate for the unknown function and its following vector and the effect of the parameter e on the convergence order are presented.In chapter two, we consider Sobolev equation:We still use characteristics mixed finite element method. In error analysis, we introduce some new thchniques to overcome difficulties in dealing with the combina-tion of the modified method of characteristics-Galerkin element procedure(MMOC-Galerkin) and the mixed finite element. Finally, we give a numerical test which shows the reliability of the new numerical method.