Augmented Finite Element Method For Nonlinear Singular Two Point Boundary Value Problems

Nonlinear singular two point boundary value problems are widely applied in many fields,such as mathematical physics,biological sciences,etc.Because of the low regularity of the solution,it is difficult to obtain the high-accuracy numerical results.For decades,several effective numerical methods have been proposed for the nonlinear singular two point boundary value problems,including difference method,spline method,collocation method,Homotopy analysis method,Adomian decomposition mathod and so on.Based on the special structure of the solution,an accurate and efficient numerical method for these problems is presented in this thesis,combining the Puiseux series decomposition method with finite element method.The main innovative idea of our method is to represent the solution by the Puiseux series expansion in a subinterval with endpoints and the singular point,containing an undetermined parameter,and then introduce the parameter associated with the singularity in the Puiseux expansion as an augmented variable.The significant advantage of the method is that the singular subinterval and regular subinterval will be connected by the augmented parameter naturally,so that a high-accuracy numerical scheme over the entire region can be obtained.Numerical experiments verify the effectiveness and accuracy of our method.The thesis is divided into six chapters.In chapter 1,we introduce the background and application of the singular two point boundary value problem,as well as some existing numerical calculation methods,briefly discuss the advantages and disadvantages of different methods,then introduce the research aim of the thesis.Chapter 2 gives the notation which used in this article and some preliminary knowledge about the Puiseux series.The third chapter focuses on the Puiseux series decomposition method based on the Green function,and gives an equivalent form for the solution and the corresponding recursive algorithm,then illustrates the existence and uniqueness of the solution represented by the Green function.Numerical examples show that the Puiseux series has local high-accuracy approximation property.In order to obtain a global high-precision approximate solution,we construct a mixed asymptotic finite element scheme in Chapter 4,which divides the entire interval into singular interval and regular interval.In the singular interval,the Puiseux series with an unknown parameter is used to approximate the exact solution,and then introduce Puiseux series with parameter as boundary condition in the regular interval to construct an augmented finite element scheme.The proposed scheme not only overcomes the difficulty that the finite element scheme cannot effectively approximate the exact solution near the singular point,but also eliminates the local property which the global approximation of the Puiseux series is poor.Numerical experiments in Chapter 5 prove that the method can solve such singular problems with high accuracy globally.Both the L2 norm and the infinite norm of the numerical solution reach the expected second-order convergence accuracy.The last chapter summarizes the whole article and proposes the further research aims and prospects.