Convergence Analysis Of Adaptive Moving Grid Methods For Sever Classes Of Singularly Perturbed Problems

Due to the existence of boundary layer or inner point layer,it is difficult to get an exact solution for singularly perturbed problem.Therefore,the research of effective numerical methods for this kind of problems has attracted the attention of scholars.This paper is based on some existing literature,the adaptive moving grid method for several classes of singularly perturbed problems is studied.The main contents are as follows:In the introduction,the research background and progress of singularly perturbed problems are introduced,and the main work of this thesis is also briefly introduced.In Chapter ?,an adaptive moving grid algorithm is discussed to solve a class of nonlinear singularly perturbed parameterized problems.At first,the backward Euler method is used to discretize the presented problems on an arbitrary nonuniform mesh,and give the corresponding local truncation error.Then,based on this local truncation error and mesh distribution principle,we prove that the semi-discretized adaptive moving grid method is first-order convergence by using the exact arc-length monitor function.Meanwhile,based on the approximation arc-length monitor function,an a posteriori error estimation is derived for the fully-discretized scheme,and a mesh generation algorithm which is easy to implement is given.At last,numerical results are given to illustrate the theoretical results.In Chapter ?,for a class of singularly perturbed Riccati problems,a upwind finite difference scheme is presented on an arbitrary nonuniform mesh,and a priori error of the discrete scheme is derived.Then,by using the equal-distribution monitor function,it is proved that the optimal order of convergence of the adaptive grid method isO(N^-1).At the same time,based on the approximate arc-length monitor function,the a posterior error estimation is given under the fully-discretized adaptive grid.Finally,numerical results are given to illustrate the theoretical results.In Chapter ?,the convergence of singularly perturbed convection diffusion equations is discussed based on upwind finite difference scheme.Firstly,the local truncation error of the discrete scheme is given on an arbitrary nonuniform mesh.Then,based on the equidistribution principle and arc-length monintor function,it is proved that the adaptive moving grid algorithm under semi-discretized scheme is first-order convergence and independent of perturbation parameters.At last,numerical results are presented to demonstrate the performance of presented method.In Chapter ?,a singularly perturbed convection-diffusion equations with Robin Boundary is considered.Firstly,the problem is transformed into a class of first-order singularly perturbed initial value system with parameters by using integral transformation,and the corresponding discrete scheme is given on an arbitrary nonuniform mesh by using the backward Euler formula.At the same time,the convergence of the semi-discretized scheme is proved by using mesh distribution principle and arc-length monitor function.Then,an a posteriori error estimation in maximum norm is derived to design an adaptive grid generation algorithm.Besides,in order to establish the parameters of the first order singularly perturbed problems,we construct a nonlinear optimization problem by using the boundary conditions,which is solved by the Nelder-Mead simplex method.At last,a linear and a nonlinear examples are presented and the numerical results are provided to demonstrate the effectiveness of our adaptive moving grid method.For these kinds of singularly perturbed problems,the convergence of the semi-discretized scheme is analyzed and an a posteriori error estimation for the fully-discretized scheme is proposed,the corresponding adaptive mesh generation algorithm is given.Furthermore,this algorithm can be extended to other singularly perturbed problems.