Multi - Scale Finite Element Numerical Simulation Of Different Boundary Of Convection - Diffusion Equation With Small Parameters

The differential equation problem with perturbed coefficients in the highest-order derivative term is called a singularly perturbed problem. If there are boundary layers phenomena, and the solution in the boundary layers will change rapidly. When we just use a uniform grid, the numerical accuracy would be low, and even if the perturbed coefficient is very small, the traditional numerical method is not a good way or even get the wrong results. Therefore in this paper, we build the priori non-uniform grids (by selecting the appropriate transition points on the boundary layer mesh). At the same time, we apply the finite element method(FEM) and multiscale finite element method(MsFEM) to solve the singular perturbation problem, respectively. This master’s thesis is divided into the following five parts:(1) In the first chapter, it is introduced for the singular perturbation problem and its research, and it outlines the necessary to address this problem.(2) In the second chapter, we describe the numerical method which is used in this paper:the multiscale finite element method. Multiscale problems and computations, multiscale finite element method and multiscale basic functions are overviewed, and the study background and basic idea of multiscale finite element are elaborated. The similarities and differences between it and traditional numerical methods are analyzed as to illustrate the advantage of multiscale finite element method.(3) In the third chapter, we provide three types of special grid structure:Shishkin grid, Bakhvalov grid and graded grid, from solutions of boundary layer in the left, right, and two sides cases, respectively for obtaining the corresponding grid node coordinates. By using these special grids we try to ensure the calculation precision.(4) In the fourth chapter, we study three different kinds of boundary conditions for the singularly perturbed convection-diffusion problem. Dirichlet boundary condition is a kind of essential condition, while for Neumann and Robin boundary condition, the derivatives of the function should be processed attentively. In this paper, we give the corresponding boundary treatment and make sure that the numerical program can be output the right simulation results.(5) In the fifth chapter, the numerical examples are given according to three different boundary conditions. According to the property of solutions, we select grid partitions and numerical methods to obtain the ideal results. Through numerical experiments we show the computational accuracy and numerical advantage of the multiscale finite element method for the small parameter singularly perturbed convection-diffusion model convincingly.