半线性抛物方程连续时空有限元方法

发布时间：2018-03-08 12:24

本文选题：抛物方程　切入点：连续时空有限元方法　出处：《内蒙古大学》2017年硕士论文　论文类型：学位论文

【摘要】：抛物方程是在工程学,物理学等实际问题中提出来的一类重要的偏微分方程.因此,数值分析和求解该类方程具有重要意义和潜在应用价值.目前学者们对偏微分方程数值解法的研究很多,大多集中在间断元,间断混合元等,但利用非标准有限元方法对偏微分方程研究的文章并不多.连续时空有限元方法统一时间和空间变量,对时间和空间两个方向同时使用有限元离散,从而在时空变量上同时发挥有限元方法的优势.本文利用了连续时空有限元方法对二维半线性偏微分方程进行了分析研究.证明了有限元解的存在唯一性并且分析抛物方程的理论误差.本文首先给出了连续时空有限元方法;在第二章中,我们得到方程在连续时空有限元空间中的有限元格式,同时定义了时空投影算子并证明了它们的性质;接下来利用Brouwer不动点定理证明了方程的连续时空有限元解的存在唯一性;最后我们通过之前所定义的时空投影算子及其性质得到了方程的理论误差估计.第三章我们给出了数值算例,验证了该方法的可行性.
[Abstract]:Parabolic equations are important partial differential equations in engineering, physics and other practical problems. Numerical analysis and solution of this kind of equations are of great significance and potential application value. At present, there are many researches on numerical solution of partial differential equations, most of which are focused on discontinuous element, discontinuous mixed element and so on. However, there are few studies on partial differential equations using nonstandard finite element methods. The continuous space-time finite element method unifies time and space variables, and discretizes time and space simultaneously in two directions: time and space. In this paper, the continuous spatio-temporal finite element method is used to analyze two-dimensional semi-linear partial differential equations, and the existence and uniqueness of finite element solutions are proved. The theoretical error of parabolic equation is analyzed. Firstly, the continuous space-time finite element method is given. In the second chapter, we obtain the finite element scheme of the equations in the continuous space-time finite element space, and define the space-time projection operators and prove their properties. Then the existence and uniqueness of the continuous space-time finite element solution of the equation are proved by using the Brouwer fixed point theorem. Finally, we obtain the theoretical error estimates of the equation by the space-time projection operator and its properties. In chapter 3, we give a numerical example to verify the feasibility of the method.
【学位授予单位】：内蒙古大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.82

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