Adaptive Method For Singularly Perturbed Problems

Adaptive grid methods are becoming established as valuable computational techniques for numerical solution of differential equations with near-singular solution. Adaptive methods are equally effective in approximation solutions of problem with boundary layers or interior layers. We mainly discuss adaptive redistributing mesh method, which is called as moving mesh method.The model of eddy viscosity and velocity distribution based on Prandtl Mixing Length in turbulent pipe flow which is source of engineering problem is solved in this, paper. It is an optimal controlling problem. The mathematical model is a nonlinear singulary perturbed differential equation with boundary layer. We discretize this nonlinear problem by using first- and second- order difference scheme including upwind scheme. We solve the discretized nonliner equations by Newton-Raphson method. The mesh is generated and moved by equidistributing principle to capture the boundary layer. We design a iterative algorithm to solve the controlling problem. The numerical result support our algorithm. Then, this paper shows how to apply the optimal mesh based on derivative error generated by interpolation function to adaptive finite method and design a efficient adaptive algorithm to move mesh based on derivative error.Our discussion is partitioned into chapters on the following topics: In the first chapter, we introduce the background and result of adaptive method; The second chapter, we solve the complicated engineering mathematical model in details; In the third chapter, we introduce the generation of optimal meshes for controlling the errors in derivative error for piecewise interpolation; In the forth chapter, we construct adaptive finite method based on derivative error; In the fifth chapter, we have some error estimation and some convergence analyses; In the end, we draw a conclusion of this work and make some comments on the prospect of adaptive method.