The Semi-Discrete Streamline Diffusion Finite Element Method For Time-Dependented Convection Diffusion Problems

The physical processes accompanied by material transport and diffusion and the flow of viscous fluid are usually mathematical model for convection-diffusion equation or the partial differential equations containing such equation. The solution of such equations often arise the local sharp(large-scale) change, such as the boundary layer and transient layer, which brings difficult to the numerical solution of such problems. Therefore, the study of the numerical solution method for convection diffusion problems is important in theoretical and practical, and can be used for environmental science, energy exploitation, fluid mechanics, electronics, and so on.The streamline diffusion finite element method for solving the convection diffusion problem is a highly efficient finite element method, which has a good numerical stability and higher precision. Previously, it used to solve time dependented convection-diffusion equation is based on the space time finite element space or finite difference finite element method. Although space time finite element method can well coordinate the space and the time flow field, and it's easy for theoretical analysis, but it paid a high price(the enormous amount of computation and storage space), especially for high-dimensional problems. And the theory analysis of the finite difference finite element is very complicated.In this paper, we give the semi-discrete streamline diffusion FEM based on the work of the predecessors. This method uses the space finite element method to discrete partial differential equation getting the ordinary differential equations of time. The ordinary differential equations of time is solved by the fourth order R-K method. We also give the theoretical analysis of the semi-discrete streamline diffusion finite element method which proves that the method is stable and convergence. And get if the interpolation precision of the finite element space is r order accuracy, and the semi-discrete streamline diffusion FEM is r—1 order convergence.In the end, we give a numerical experiments for the problem with a boundary layer. We compare the numerical results to the FDSD's[l], that proves the algorithm is feasible and effective, and has a high numerical precision.