具有长条形内边界的二维调和外问题的基于自然边界归化的数值方法

【作者】 黄红英；

【导师】 杜其奎；

【作者基本信息】 南京师范大学 ， 计算数学， 2004， 硕士

【摘要】 由冯康先生首创并发展起来的自然边界元与有限元、辛几何算法一起构成了冯先生的三大学术贡献。后经余德浩教授等人的进一步发展，除了自然边界元法可以直接用来求解某些特殊区域上的椭圆边值问题以外，自然边界元与有限元耦合法、基于自然边界归化的区域分解算法等还是处理无界区域及断裂区域问题的有效手段。前述研究工作已在二维、三维领域以及与时间有关双曲型、抛物型问题方面取得了许多重要研究成果。但这些成果主要是基于圆周(或球面)人工边界，而对于具有长条形内边界的无界区域问题应用椭圆(或椭球面)人工边界可大大减少计算量，节省存储空间和计算时间，不失为一种更有效的数值方法。 本文借助于区域分解算法的思想并基于椭圆人工边界的自然边界归化理论，研究具有长条形内边界的二维调和外问题的自然边界元与有限元的耦合法及一种非重叠型区域分解算法。第一章提出了具有长条形内边界的二维调和外问题的自然边界元与有限元的耦合法，讨论耦合变分问题的适定性，并给出了逼近解的误差估计及数值例子。第二章提出基于椭圆人工边界的非重叠型区域分解算法(即D-N交替算法)，研究一般外区域上的D-N交替算法离散问题的收敛性，证明其收敛速度与有限元网格粗细无关，详细分析了椭圆外区域上的D-N交替算法的收敛性与松弛因子的选取，并给出相应的数值例子。更多还原

【Abstract】 The natural boundary reduction, first suggested and developed by Prof.K.Feng, is the third academic contribution of Prof.Feng besides the finite element method and the sympletic algorithms. Later Prof.Yu has done a lot of outstanding work in the field. Now not only can the NBEM(natural boundary element method) be used to deal with some boundary value problems over several special domains directly, but also the DDM(domain decomposition method) and the coupled algorithm, based on the natural boundary reduction, are two kinds of efficient methods in solving problems over unbounded or cracked domains. Up to now, the above work has achieved many important results for 2-D problems, 3-D problems and time-dependent problems successfully. However all of them are done based on an circular or a spherical artificial boundary. As for some boundary value problems over an unbounded domain outside an elongated domain, computing number will be reduced largely and storage space and computing time will be saved with an elliptic or an ellipsoidal artificial boundary.In this paper, through the idea of domain decomposition method and the theory of natural boundary reduction, the coupling of natural boundary element and finite element methods and a non-overlapping domain decomposition method based on natural boundary reduction, are investigated for 2-D exterior harmonic problem outside an elongated domain. In the first chapter, we propose the coupling of NDEM and FEM, discuss well-posedness of the coupled variational problem , and give error estimates of approximate solutions and some numerical examples. In the second chapter, a non-overlapping domain decomposition method (i.e. D-N alternating algorithm) based on natural boundary reduction on an elliptic artificial boundary is suggested for 2-D exterior harmonic problem outside an elongated domain. The convergence of discrete D-N alternating algorithm for a general exterior domain is investigated, the convergence rate independent of the mesh size is proved, the convergence of the D-N alternating algorithm and the choice of the contraction factor for an exterior elliptic domain are analyzed, and some numerical results are given.更多还原