Spline Function For Solving The Nonlinear Differential Equations

With the rapid development of computer technology, spline function is widely applied in numerical approximation, CAGD (Computer-Aided Geometric De-sign), numerical solutions of differential equations and becomes an efficient math-ematical tool for engineers. Spline function also plays an important role in the finite element theory. It is known that, the finite element space is a special spline space actually [40]. The choice of the spline space is related to the efficiency of the algorithm, and is the key to success of the finite element method [81]. More and more attentions are payed on the collocation methods and finite difference schemes based on the spline function[56].The main purpose of this thesis is to discuss the spline method and its appli-cation in numerical solutions of the nonlinear differential equations. As far as we know, the univariate B-spline and multivariate spline B-form(B-net) are always one's first choice, owing to their outstanding performance [8,49]. On the other hand, the well-defined properties of the constructive spline function bring great conveniences to the numerical analysis and application, such as, easy to represent and efficient to evaluate. Therefore, there are many literatures on B-spline for solving the differential equations. Recently, Ming-Jun Lai et al. [50] apply the multivariate spline B-form to multivariate function fit and approximation and to the numerical solutions of partial differential equations (linear partial differential equations). Some efficient, and stable algorithms for evaluating their derivatives and integral are presented by using the B-form.The thesis is organized as follows. In chapter 1, we give a brief introduction to the history of spline function. For univariate spline, we mainly discuss some important properties and the applications of the B-spline. We shall pay more attentions to the B-form of bivariate spline, and basic algorithms with respect to the B-form, e.g., de Casteljau algorithm which plays an important role in the computation of derivatives and integrals, etc. The conforming/smoothness condition of B-form on adjacent two triangles is the basis of the multivariate spline function theory. By assembling the boundary conditions and the smooth-ness condition into a global matrix, we may discretize the original systems into saddle point problems which are easy to implement. The degree raise theorem presented in this chapter is the basis of the local p-adaptivity and the p-version two-level algorithm. We also give a simple description of the weak solution of the differential equation, the mathematical theory of the finite element method. Green function theory is introduced, since it is a useful tool in researching the differential equation. It is employed to proof the convergence of the nonlinear iteration in the preceding chapter. At the last of this chapter, we explain the Green function in a physical model.In chapter 2, we present high-accurate collocation methods based on n-th B-spline basis for solving non-linear singular boundary value problems. The reg-ularity of the singular linear operator L0u≡-u"(x)-α/xu'(x) is discussed and investigated. Namely, when u satisfies u'(0) = 0,u(1)= 0, we have c1‖u"‖≤‖L0u‖≤c2‖u"‖. It is proved that the iterative algorithm converges to a smooth approximate solution of problems monotonely, provided the algorithm is applied tion method, collocation method based on variational and least square method are all developed and compared as computational methods.Convergence analy-sis is established through Green function theory and fix points theory. Finally, we give some numerical illustrations to demonstrate the efficiency of the B-spline function method.In chapter 3, we apply the bivariate spline in B-form to solve the nonlinear elliptic boundary value problems. Similar to the case of univariate, maximum principle and Green function theory provide the convergence of the Picard itera-tion series. Given the spline function space, we discretize the variational problems which are equivalent to the original systems. Subjected to the smoothness con-ditions of B-form and the boundary conditions, the system after discretization is changed into a saddle point problem. An effective iterative algorithm for solving these saddle problems is introduced. We prove that, the Newton iteration series of the variational problem possess second order convergence. The matrix form of Gaussian quadrature reduces the complexity of numerical integration of nonlin- ear function. The h-adaptivity and p-adaptivity spline methods are introduced and applied to solve the nonlinear partial differential equations. More detail on the adaptive spline method for Poisson equation can be referred to [53]. However, the posteriori error estimator is quite different from [53], due to the existence of the nonlinear term f(x,u).Nevertheless, the solution of the nonlinear equation can be still quite com-putationally intensive, especially when the mesh size is tiny or the degree is high. Two-level algorithm is presently a very promising approach for solving these problems. The two-level method based on the mesh refinement (h-version) [91] is also called two-grid method in some literatures. In chapter 4, we consider the "p-version" two-level method. B-coefficients of the degree-raised polynomial provide a good support for the implementation of these algorithms. According to the Ref. [92], we introduce the p-version two-level method based on Picard iteration, mixed iteration, Newton iteration and modified Newton iteration, re-spectively. The error estimate and convergence analysis are established. The scaling between the subspace and the fine space should be determined to ob-tain the asymptotically optimal accuracy. Numerical example is given to show the convergence of these algorithms. It can be seen from the numerical results that p-version two-level method is an attractive method for solving the nonlinear problem, and saves the CPU-time considerably.In chapter 5, we consider the application of p-version two-level methods for the stream function form of the Navier-Stokes equations. In [51], Lai and Wenston discuss the numerical approximations of the NS equation using bivariate splines of arbitrary degree d and smoothness r with d≥3r+2. However, the high accuracy method costs a lot of CPU time, especially when mesh size h is small and the degree d is large. Therefore, we apply two algorithms of p-version two-level method to solve the system. The two algorithms are based on the Picard iteration and Newton iteration respectively. In this chapter, we also apply the p-version two-level based on modified Newton iteration to the NS equations and analyze the convergence order. Matrix forms of the algorithms are given, respectively. Finally, we present some simple numerical results to demonstrate the efficiency of our proposed schemes.