几类问题基于自然边界归化的算法研究
发布时间：2018-09-04 05:32
【摘要】：自然边界归化理论是由冯康教授首创,经由其及余德浩教授等学者发展完善.该理论与有限元方法和辛几何算法构成了冯先生的主要学术贡献.自然边界元法,可以直接用于求解某些无界区域椭圆边值问题,其与有限元方法、区域分解算法和多重网格算法的耦合算法亦是处理无界区域及凹角、断裂区域问题的有效手段之一,并在二维及三维领域内取得了许多重要的研究成果.之前的研究通常以圆(二维情况)、球面(三维情况)作为人工边界,但对于某些特殊区域,例如,长条型区域,用长椭球面或椭圆作人工边界,则可大大减小计算区域,从而可以减少计算量和存储量.本文主要研究三维各向异性外问题的基于椭球面人工边界的区域分解算法(Schwarz交替算法和D-N交替算法)和二维Helmholtz方程外问题的多重网格算法.第一章介绍了两类正交坐标系、几类特殊函数和Sobolev空间的相关概念和定理,作为以后各章进行理论分析的重要工具.第二章研究了三维各向异性外问题的基于基于自然边界归化的Schwarz交替算法.首先对所研究问题进行变量替换,得到相应的Laplace方程外问题.进一步得到旋转椭球外区域上问题自然积分方程和Poisson积分公式.然后,给出Schwarz交替算法,分析了该算法的收敛性,并给出了数值解的误差估计,通过数值算例以示算法的可行性与有效性.第三章讨论了三维各向异性外问题的基于旋转球面人工边界的D-N交替算法.根据第二章相应内容,给出D-N交替算法和等价的Richardson迭代算法.其次,分析了该算法的收敛性,给出了等价的变分形式及其离散形式.然后,对其离散形式进行收敛性分析,最后通过数值算例以示方法的可行性与有效性.第四章研究了二维Helmholtz方程外问题的基于自然边界元方法的多重网格算法.首先给出了问题等价的变分形式,其次建立了多重网格算法并分析了该算法的收敛性、收敛速度分析及离散情形的误差估计.最后,通过数值算例以示方法的可行性与有效性.
[Abstract]:Naturalization theory of natural boundary was initiated by Professor Feng Kang and developed and perfected by Professor Yu Dehao and other scholars. The theory, the finite element method and the symplectic geometric algorithm constitute Mr. Feng's main academic contribution. The natural boundary element method can be directly used to solve the elliptic boundary value problems in some unbounded regions. The coupling algorithms of the natural boundary element method and the finite element method, the domain decomposition algorithm and the multi-grid algorithm are also used to deal with the unbounded region and the concave angle. One of the effective methods of fault region problem, many important research results have been obtained in two and three dimensions. Previous studies usually use circles (two-dimensional cases) and spherical surfaces (three-dimensional cases) as artificial boundaries, but for some special regions, such as long ellipsoid or ellipse, the computational area can be greatly reduced. Thus, the amount of computation and memory can be reduced. In this paper, the domain decomposition algorithm based on ellipsoidal artificial boundary (Schwarz alternating algorithm and D-N alternating algorithm) and the multigrid algorithm for two-dimensional Helmholtz equation are studied. The first chapter introduces two kinds of orthogonal coordinate systems, some special functions and some related concepts and theorems of Sobolev space, which are important tools for theoretical analysis in the following chapters. In chapter 2, we study the Schwarz alternating algorithm based on natural boundary normalization for three dimensional anisotropic exterior problems. First, the variables are replaced to obtain the corresponding problem of Laplace equation. Furthermore, the natural integral equation and Poisson integral formula for the problem in the outer domain of a rotating ellipsoid are obtained. Then, the Schwarz alternating algorithm is given, the convergence of the algorithm is analyzed, and the error estimation of the numerical solution is given. A numerical example is given to show the feasibility and effectiveness of the algorithm. In chapter 3, the D-N alternating algorithm based on the artificial boundary of rotating sphere is discussed. According to the corresponding contents of the second chapter, D-N alternating algorithm and equivalent Richardson iterative algorithm are given. Secondly, the convergence of the algorithm is analyzed, and the equivalent variational form and its discrete form are given. Then, the convergence of the discrete form is analyzed. Finally, numerical examples are given to show the feasibility and effectiveness of the method. In chapter 4, the multi-grid algorithm based on natural boundary element method for two-dimensional Helmholtz equation is studied. In this paper, the equivalent variational form of the problem is given, and then the multigrid algorithm is established and its convergence, convergence rate analysis and error estimation in discrete case are analyzed. Finally, numerical examples are given to demonstrate the feasibility and effectiveness of the method.
【学位授予单位】：南京师范大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O241.8
本文编号：2221161
[Abstract]:Naturalization theory of natural boundary was initiated by Professor Feng Kang and developed and perfected by Professor Yu Dehao and other scholars. The theory, the finite element method and the symplectic geometric algorithm constitute Mr. Feng's main academic contribution. The natural boundary element method can be directly used to solve the elliptic boundary value problems in some unbounded regions. The coupling algorithms of the natural boundary element method and the finite element method, the domain decomposition algorithm and the multi-grid algorithm are also used to deal with the unbounded region and the concave angle. One of the effective methods of fault region problem, many important research results have been obtained in two and three dimensions. Previous studies usually use circles (two-dimensional cases) and spherical surfaces (three-dimensional cases) as artificial boundaries, but for some special regions, such as long ellipsoid or ellipse, the computational area can be greatly reduced. Thus, the amount of computation and memory can be reduced. In this paper, the domain decomposition algorithm based on ellipsoidal artificial boundary (Schwarz alternating algorithm and D-N alternating algorithm) and the multigrid algorithm for two-dimensional Helmholtz equation are studied. The first chapter introduces two kinds of orthogonal coordinate systems, some special functions and some related concepts and theorems of Sobolev space, which are important tools for theoretical analysis in the following chapters. In chapter 2, we study the Schwarz alternating algorithm based on natural boundary normalization for three dimensional anisotropic exterior problems. First, the variables are replaced to obtain the corresponding problem of Laplace equation. Furthermore, the natural integral equation and Poisson integral formula for the problem in the outer domain of a rotating ellipsoid are obtained. Then, the Schwarz alternating algorithm is given, the convergence of the algorithm is analyzed, and the error estimation of the numerical solution is given. A numerical example is given to show the feasibility and effectiveness of the algorithm. In chapter 3, the D-N alternating algorithm based on the artificial boundary of rotating sphere is discussed. According to the corresponding contents of the second chapter, D-N alternating algorithm and equivalent Richardson iterative algorithm are given. Secondly, the convergence of the algorithm is analyzed, and the equivalent variational form and its discrete form are given. Then, the convergence of the discrete form is analyzed. Finally, numerical examples are given to show the feasibility and effectiveness of the method. In chapter 4, the multi-grid algorithm based on natural boundary element method for two-dimensional Helmholtz equation is studied. In this paper, the equivalent variational form of the problem is given, and then the multigrid algorithm is established and its convergence, convergence rate analysis and error estimation in discrete case are analyzed. Finally, numerical examples are given to demonstrate the feasibility and effectiveness of the method.
【学位授予单位】：南京师范大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O241.8
【参考文献】
相关期刊论文 前3条
1 余德浩;无界区域上基于自然边界归化的一种区域分解算法[J];计算数学;1994年04期
2 余德浩;断裂及凹角扇形域上调和正则积分方程的数值解[J];数值计算与计算机应用;1983年03期
3 冯康;;论微分与积分方程以及有限与无限元[J];计算数学;1980年01期
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