Multi - Scale Finite Element Method For Solving Singularly Perturbed Problems By Adaptive Grid

In scientific computing, we often encounter with differential equation with small parameter, which is called singularly perturbed problem. The solution of this problem changes rapidly in the local boundary layers, so its numerical accuracy on a uniform grid is very low. While the adaptive grid without altering the total number of nodes, it can distribute more nodes to the rapidly changing area, so that the numerical solution may increase its accuracy greatly.In this paper, we study the singular perturbation on the adaptive grids in one dimensional space, and combine the finite element and multiscale finite element method to make the accuracy much better.At present, numerical algorithms for solving the singular perturbation are mainly divided into two kinds, one is operator adaptive method, and the other is grid adaptive method. The main work of this paper is about the adaptive grid to solve the singularly perturbed problem. We introduce several special grids, and the definition of grid nodes are given, and we carry on the numerical computation for different singularly perturbed equation on uniform grid, Shishkin grid, Graded grid, Bakhvalov grid, respectively. Through the analysis of the numerical results, we find the similar conclusion:on the uniform grid generally we can’t get satisfactory numerical solution, while on the adaptive grid our numerical solution improves much better even in the boundary layer. It’s shown that the adaptive grid combining with multiscale finite element method for solving the singular perturbation we obtain a good performance really, and it can be extended to more general or complex cases. Through many computations and figures, we demonstrate the effective approximation between the numerical solution and the exact solution, and list the corresponding results and convergent order. The multiscale finite element method is more efficient than the traditional finite element method, since it executes on the coarse grid to save more computer resource. In this way, the multiscale method has many advantages in the large-scale numerical calculation, and it is more adapted to the big data era. Then we consider the case without exact solution for singularly perturbed equation with variable coefficients, and take the finite element solution on the uniform fine partition 2048 as an exact reference.Comparing with the multiscale solution on the relative coarse grid we demonstrate its efficiency, and present the relevant results and scientific facts.In the last chapter of this paper, we consider one dimensional non-stationary convection diffusion equation depending on the time, and outline the relevant treatment and idea. But so far from now, there is no detailed numerical experiment, we wish to carry out a more in-depth research in the future.