具有周期初始条件的微分方程间断有限元法研究

发布时间：2018-05-30 20:35

本文选题：周期初始条件 + 一维双曲微分方程　；参考：《湖南科技大学》2017年硕士论文

【摘要】：间断有限元方法是使用完全不连续的分片多项式空间作为解空间和检验函数空间的一类有限元方法,间断有限元法解偏微分方程的超收敛性质也是最近几年来本研究领域学者们非常感兴趣的研究主题。本文研究了求解一类一维具有周期初始条件的微分方程间断有限元计算方法及其收敛性质。本文主要研究了具有周期初始条件的一阶双曲微分方程和抛物方程定解问题。对于一般的一阶双曲方程,首先将其转化等价具有周期边界的混合边界问题,研究了选取迎风数值流量时对应的有限元方法,构造校正函数得到超逼近有限元的插值函数,依次证明了一次间断有限元和任意间断有限元的逐点误差以及区间平均值误差估计;其次推导了一次有限元的时间向前全离散计算格式和向后全离散计算格式、二次有限元的4阶Runge-Kutta全离散计算格式;最初给出了两个数值例子验证了计算方法的有效性。对于一般的抛物方程定解问题,简单介绍了局部间断元方法,并推导了一次元的时间向前全离散计算格式和向后全离散计算格式;二次有限元的4阶Runge-Kutta全离散计算格式。
[Abstract]:Discontinuous finite element method is a kind of finite element method which uses completely discontinuous piecewise polynomial space as solution space and test function space. The superconvergence property of discontinuous finite element method for solving partial differential equations is also a subject of great interest to scholars in this field in recent years. In this paper, the discontinuous finite element method for solving a class of one-dimensional differential equations with periodic initial conditions and its convergence properties are studied. In this paper, we study the solutions of first order hyperbolic differential equations and parabolic equations with periodic initial conditions. For a general first order hyperbolic equation, the mixed boundary problem with periodic boundary is transformed into a mixed boundary problem. The finite element method corresponding to the selection of upwind numerical flux is studied, and a correction function is constructed to obtain the interpolation function of the superapproximate finite element. The point-by-point error and interval mean error estimation of one-order discontinuous finite element and arbitrary discontinuous finite element are proved in turn. The fourth order Runge-Kutta full discrete scheme of quadratic finite element method is presented, and two numerical examples are given to verify the validity of the method. For general parabolic equations, the local discontinuous element method is briefly introduced, and the time forward and backward full discrete schemes of the first order element and the fourth order Runge-Kutta full discrete scheme of quadratic finite element are derived.
【学位授予单位】：湖南科技大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.82

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