偏微分方程的耦合Legendre Chebyshev配置最小二乘法

发布时间：2018-04-20 05:42

本文选题：变系数 + 椭圆方程　；参考：《上海大学》2016年博士论文

【摘要】：科学与工程计算中,有限差分方法,有限元方法,以及有限体积方法都是解偏微分方程的有效数值方法.近年来,最小二乘有限元法也引起了部分学者的关注和发展.为了有高阶精度,有学者还考虑了最小二乘谱方法.本文主要研究偏微分方程的Legendre-Galerkin Chebyshev配置(LGCC)最小二乘法.即,通过引进一个通量将原问题写成等价的一阶系统,然后对其建立LGCC最小二乘方法.接着,对该格式进行理论分析以及用数值算例加以验证.本文内容可归纳为如下几部分：第一部分考虑一维的偏微分方程.首先,针对变系数两点边值问题构造LGCC最小二乘法.接着推导格式的强制性和连续性,以及误差估计.其次,还发展该方法的多区域形式,并探讨算法的并行化和推广单区域的理论结果.然后,研究含两类非齐次跳跃条件抛物方程的多区域LGCC方法,该格式对第一类和第二类跳跃条件分别本性和自然性处理.该方法被用于Stefan问题的计算.最后,考虑含间断变系数两点边值问题的多区域LGCC最小二乘格式和对应的理论分析.第二部分对两维的椭圆方程,首先,研究两维变系数椭圆方程的LGCC最小二乘法.给出两维Chebyshev插值算子的稳定性分析和逼近结论,由此导出格式的强制性和连续性,以及得到关于H1-范数的误差估计.其次,将该方法应用于Stokes方程的求解以及给出其理论分析.然后,发展含两类跳跃条件两维变系数椭圆方程的多区域LGCC以及对其设计并行实施算法.最后,还考虑含跳跃条件变系数椭圆方程的多区域LGCC最小二乘法以及探讨其并行实施算法.推广前面对光滑函数的Chebyshev插值的稳定性分析和相关的理论结论.第三部分针对三角形上变系数椭圆方程,我们发展Legendre-Galerkin数值积分(LG-NI)三角单元最小二乘法,并且对其强制性与连续性,以及收敛性进行分析.进一步,结合对区域内部使用矩形单元划分和对边界采用适当三角元剖分,还研究多边形上变系数椭圆方程的多区域LG-NI最小二乘法和它的收敛性.最后一部分探讨发展方程LGCC最小二乘法.首先构造抛物方程LGCC最小二乘法以及其的多步形式,并且给出它在Burgers的应用.进一步,讨论两种非线性发展方程的LGCC最小二乘法.对Burgers方程,先对它的一阶系统用Crank-Nicolson方法离散,接着对其空间设计LGCC最小二乘法.另外,将该方法应用于两维非线性抛物方程.本文的方法基于Legendre-Galerkin最小二乘法,对变系数用Chebyshev插值处理.该方法可导出对称正定代数方程,使其便于应用迭代方法求解.注意到它还继承了Legendre的良好稳定性和避免Chebyshev权函数在区域的交界出现的奇性。
[Abstract]:In scientific and engineering calculation, finite difference method, finite element method and finite volume method are all effective numerical methods for solving partial differential equations. In recent years, the least square finite element method has also attracted the attention and development of some scholars. In order to have higher order accuracy, some scholars also consider the least square spectral method. In this paper, the Legendre-Galerkin Chebyshev collocation least square method for partial differential equations is studied. That is, by introducing a flux, the original problem is written as an equivalent first-order system, and then the LGCC least squares method is established for it. Then, the scheme is theoretically analyzed and verified by numerical examples. The content of this paper can be summarized as follows: the first part considers one-dimensional partial differential equation. Firstly, the LGCC least square method is constructed for the two point boundary value problem with variable coefficients. Then the mandatory and continuity of the format and error estimation are derived. Secondly, the multi-region form of the method is developed, and the parallelization of the algorithm and the theoretical results of extending the single region are discussed. Then, the multiregion LGCC method with two classes of nonhomogeneous jump conditions is studied. The scheme deals with the nature and naturalness of the first and second kinds of jump conditions, respectively. The method is used to calculate the Stefan problem. Finally, the multi-region LGCC least-squares scheme with discontinuous variable coefficient two-point boundary value problem and its corresponding theoretical analysis are considered. In the second part, the LGCC least square method for the two-dimensional elliptic equation with variable coefficients is studied. In this paper, the stability analysis and approximation results of two-dimensional C hebyshev interpolation operator are given. The mandatory and continuity of the scheme are derived, and the error estimates about H _ 1-norm are obtained. Secondly, the method is applied to the solution of Stokes equation and its theoretical analysis is given. Then, the multi-domain LGCC with two kinds of jump conditions for the elliptic equations with variable coefficients is developed and its parallel implementation algorithm is designed. Finally, the multiregion LGCC least square method with variable coefficient elliptic equations with jump conditions is considered and its parallel implementation algorithm is discussed. The stability analysis of Chebyshev interpolation for smooth functions and related theoretical conclusions are generalized. In the third part, for the elliptic equation with variable coefficients on the triangle, we develop the Legendre-Galerkin numerical integral and LG-NI-based least square method, and analyze its compulsion, continuity and convergence. Furthermore, the multi-region LG-NI least square method and its convergence for elliptic equations with variable coefficients on polygon are studied by using rectangular element partitioning and triangulation of boundary. In the last part, the development equation LGCC least square method is discussed. First, the parabolic equation LGCC least square method and its multistep form are constructed, and its application in Burgers is given. Furthermore, the LGCC least square method for two nonlinear evolution equations is discussed. For the Burgers equation, the first order system is discretized by Crank-Nicolson method, and then the LGCC least square method is designed for its space. In addition, the method is applied to two dimensional nonlinear parabolic equations. The method of this paper is based on Legendre-Galerkin least square method, and the variable coefficient is treated by Chebyshev interpolation. This method can be used to derive symmetric positive definite algebraic equations and make them easy to be solved by iterative method. It also inherits the good stability of Legendre and avoids the singularity of Chebyshev weight function at the junction of region.
【学位授予单位】：上海大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O241.82

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