间断Galerkin有限元方法的误差估计与超收敛性

发布时间：2018-06-13 23:32

本文选题：间断Galerkin有限元 + 误差估计　；参考：《哈尔滨理工大学》2017年硕士论文

【摘要】：间断Galerkin有限元方法是一类使用允许完全间断多项式空间作为有限元空间的高分辨率数值方法。因其具有可以达到任意阶精度、容易实现h-p自适应和能够处理复杂计算区域等优点,在工程计算方面有着广泛的应用,所以对间断Galerkin有限元方法的理论研究具有十分重要的意义。本文研究的三种发展型偏微分方程分别是变系数的线性双曲守恒律方程、非线性对流扩散方程和线性五阶偏微分方程。主要讨论了这三种方程的间断Galerkin有限元解的误差估计和超收敛性。针对变系数的线性双曲守恒律方程,通过有限元分析的方法和构造特殊的投影给出了使用迎风型数值通量的间断Galerkin有限元方法的误差估计。本章最后的数值试验部分不仅验证了收敛阶是最优的,还观察到相比于方程真解,数值解更加接近真解特殊投影这一超收敛性质。为进一步研究方程系数变号时的超收敛性指明了方向。而对于非线性对流扩散方程,证明了对流项导数保号时,间断Galerkin有限元解以高于误差收敛阶半阶的速度超收敛于方程真解的Gauss-Radau投影。几个算例也印证了理论的正确性,最后通过构造误差指示器展示了超收敛性的应用。最后讨论了线性五阶方程局部间断Galerkin有限元格式的误差估计和超收敛性。本章采用了新思路选取试验函数,得到了时间上线性增长的误差和超收敛形式,说明局部间断Galerkin有限元的数值解和真解的误差在很长的时间内都不会显著增长,体现了局部间断Galerkin有限元方法求长时间数值解时的优越性。不过,通过数值试验发现,本章的超收敛阶不是最优的。
[Abstract]:The discontinuous Galerkin finite element method is a kind of high resolution numerical method which uses the fully discontinuous polynomial space as the finite element space. Because of its advantages of achieving arbitrary order accuracy, easy realization of h-p adaptation and ability to handle complex computing areas, it is widely used in engineering calculation. Therefore, the theoretical study of discontinuous Galerkin finite element method is of great significance. The three kinds of evolution partial differential equations studied in this paper are linear hyperbolic conservation law equation with variable coefficients, nonlinear convection-diffusion equation and linear fifth order partial differential equation. The error estimates and superconvergence of discontinuous Galerkin finite element solutions for these three equations are discussed. For the linear hyperbolic conservation law equation with variable coefficients, the error estimates of the discontinuous Galerkin finite element method using upwind numerical flux are given by means of finite element analysis and special projection. In the last part of this chapter, not only the convergence order is proved to be optimal, but also the superconvergence property that the numerical solution is closer to the special projection of the true solution than the true solution of the equation is observed. The direction of superconvergence is pointed out for the further study of the superconvergence when the coefficients of the equation are changed. For the nonlinear convection-diffusion equation, it is proved that the discontinuous Galerkin finite element solution superconverges to the Gauss-Radau projection of the true solution of the equation at a speed higher than the half-order of error convergence when the convection term derivative is signed. Several examples also prove the correctness of the theory. Finally, the application of superconvergence is demonstrated by constructing error indicator. Finally, the error estimates and superconvergence of locally discontinuous Galerkin finite element schemes for linear fifth order equations are discussed. In this chapter, a new idea is adopted to select the test function, and the error and superconvergence form of linear growth in time are obtained. It is shown that the error of numerical solution and true solution of locally discontinuous Galerkin finite element does not increase significantly in a very long period of time. The superiority of the local discontinuous Galerkin finite element method in solving the numerical solution of long time is demonstrated. However, numerical experiments show that the superconvergence order of this chapter is not optimal.
【学位授予单位】：哈尔滨理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.82

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