Application Of C-Bézier And H-Bézier Spline Finite Element Method In Solving Parabolic Equations

The finite element method is an important tool for solving partial differential equations in various fields of modern science and engineering.Although the accuracy of numerical solutions can be improved by proper mesh refinement,the computational complexity will also increase.It’s an essential topic in finite element method to improve the accuracy of numerical solutions.Spline functions have a pivotal role in data fitting,function approximation and numerical solution of partial differential equations,etc.In this thesis,a numerical algorithm combining spline functions with the finite element method is developed to solve the heat conduction equations and convection-diffusion problems,and the stability and error estimation of discrete scheme are analyzed.The main contents are as follows:In Chapter 2,two kinds of spline functions,C-Bezier basis function and HBezier basis function,proposed by Wang et al.,are recalled.These basis functions are generalizations of the classical Bernstein basis,which not only inherit most of their geometric properties,but also can accurately represent high-order polynomial curves and conics.Furthermore,the C-Bezier and H-Bezier basis functions have a free shape parameter which provides a lot of convenience and flexibility to the simulation of numerical solutions.In Chapter 3,the error of θ-difference finite element method for parabolic equations is analyzed.Since the stability of the numerical solutions is related with the accuracy of the initial value,this chapter discuss the stability and error estimation of θ-difference scheme which is the generalization of back Euler method,forward Euler method and improved Euler method with the properties of interpolation and projection operators.In Chapter 4,the finite element method based on C-Bezier and H-Bezier basis function is established to solve the heat conduction equations.Firstly,the function spaces of the finite element method are constructed using C-Bezier and H-Bezier basis functions,and the semi-discrete scheme of the heat conduction equations is obtained by Galerkin finite element method.Secondly,the fully discretized Galerkin method is obtained by further discretizing time variable applying θ-difference scheme.Finally,by considering several numerical examples,the accuracy of approximate solutions is improved by 1-4 order-of magnitudes compared with Lagrange basis function,it is shown that these two kinds of basis functions have better approximations in simulating heat conduction problems.In Chapter 5,the convection-diffusion equations are solved numerically by the finite element method with C-Bézier or H-Bézier basis functions.Firstly,we apply CBézier and H-Bézier basis functions to constructed the test and trial function spaces of the finite element method.Secondly,the θ-differential finite element method is used to discrete the time variable of the convection diffusion equations,and a fully discretized Galerkin finite element method is obtained.Finally,several examples are presented to verify the feasibility and effectiveness of our method.Compared with Lagrange basis functions,numerical accuracy is improved by 1-3 order-of magnitudes which implied a much better approximation in simulating convectiondiffusion problems.