无界区域抛物方程自然边界元方法

本文应用自然边界元方法求解无界区域抛物型初边值问题。首先将控制方程对时间进行离散化,得到关于时间步长离散化的椭圆型问题。通过Fourier展开,导出相应问题的自然积分方程和Poisson积分公式。研究了自然积分算子的性质,并讨论了自然积分方程的数值解法,最后给出数值例子。从而解决了抛物型问题的自然边界归化和自然边界元方法。     

一种新的局部时间积分的区域分解波形松弛算法

提出一种新的区域分解波形松弛算法, 使得可以在不同的子域采用不同的时间步长来并行求解线性抛物方程的初边值问题. 与传统的区域分解波形松弛算法相比, 该算法可以通过预条件子来加快收敛速度, 并且对内存的需求大大降低. 给出了局部时间步长一种具体的实现方法, 证明了离散解的存在唯一性, 并在时间连续水平分析了预条件系统. 数值实验显示了新算法的有效性.     

无界区域Stokes 问题非重叠型区域分解算法及其收敛性

本文研究无界区域Stokes方程外问题的利用有限元法和自然边界归化的非蕈叠型区域分解算法,此方法对无界区域Stokes问题非常有效.给出连续和离散情形的D-N算法及其收敛性分析,得到算法收敛的充要条件及充分条件,并得到最优的松弛因子和压缩因子,最后给出数值算例予以验证.     

凹角区域双曲外问题的精确人工边界条件

研究一类凹角区域双曲型外问题的数值方法.先用Newmark方法对时间进行离散化,在每个时间步求解一个椭圆外问题.然后引入人工边界,并获得精确的人工边界条件.给出半离散化问题的变分问题,证明了变分问题的适定性,并给出了误差估计.最后给出数值例子,以示该方法的可行性与有效性.     

椭圆外区域各向异性问题基于人工边界条件的Schwarz交替算法

研究一类二维各向异性外问题的重叠型区域分解.基于自然边界归化,对各向异性外问题提出了一种Schwarz交替算法,并给出其离散形式,分析了算法的收敛性.给出数值试验以示算法的可行性与有效性.     

求解二维半线性抛物方程的校正型显隐区域分解算法

本文研究了利用分布式并行计算系统求解二维半线性抛物方程的内边界校正型显隐区域分解（CEIDD）算法．在实际问题中通常利用简洁的直线内边界（sI）将空间区域分解成若干个相互不重叠的条状或块状子区域．利用Leray-Schauder不动点定理和离散能量方法证明了基于不交叉直线内边界的CEIDD—SI算法的唯一可解性,无条件稳定性和收敛性,并得到了一个改进的误差估计．当直线内边界在区域内部相互交叉时,这种在内边界上追加了隐式校正步的算法需要在每一个时间层进行全局通信,从而使算法的并行可扩展性大为降低．为克服这一缺点,设计了一种由直线和锯齿形接点组合而成的复合内边界（CI）．分析表明,基于复合内边界的CEIDD—CI算法无条件稳定、通信效率高、可以直接利用现有的串行算法计算子区域的隐式解,是一类可扩展的并行算法．为验证算法的稳定性和收敛性,文中给出了两个具体算例．     

单调迭代结合虚拟区域法求解非线性障碍问题

讨论了二阶半线性椭圆方程障碍问题的数值求解问题.用单调迭代算法求解障碍问题,并用改进的虚拟区域法求解相关的不规则区域上具有Dirichlet边界条件的椭圆方程.在计算过程中,传统的有限元离散会导致用扩展区域规则网格计算不规则物体边界上积分的困难.为了克服此困难,给出了一种新的基于有限差分的算法,从而使得偏微分快速算法可用.算法结构简单,易于编程实现.对有扩散和增长障碍的logistic人口模型数值模拟说明算法可行且高效.     

一类三维变动区域上的非线性抛物型方程之全离散有限元形式

1 引言 1986年,L.Cermak和M.Zlamal研究了半导体器件中杂质的重新分布,对具有活动边界的二维非线性扩散问题。给出在时间方向上是一阶精度的全离散有限元格式。证明了格式最优的H~1模和次最优的L~2模估计。1989年.P.Lesaint和R.Touzani对一维变动区域上的热传导方程。经过坐标变换,给出了在固定区域上的全离散有限元格式和最优的L~2模估计。1990年,梁国平和陈志明利用时空有限元,给出了变动区域上线性抛物型的方程的全离散变网格有限元格式。证明了最优的L~2收敛性。本文考虑了一类具有活动边界的三维     

椭圆外区域上的自然边界元法

１．引言 二十年来,自然边界元法已在椭圆问题求解方面取得了许多研究成果。它可以直接用来解决圆内（外）区域、扇形区域、球内（外）区域及半平面区域等特殊区域上的椭圆边值问题［１,２,５］,也可以结合有限元法求解一般区域上的椭圆边值问题,例如基于自然边界归化的耦合算法及区域分解算法就是处理断裂区域问题及外问题的一种有效手段［２－４,６］。 人们在设计求解外问题的耦合算法或者区域分解算法时,通常选取圆周或球面作人工边界。但对具有长条型内边界的外问题,以圆周或球面作人工边界显然并非最佳选择,它将会导致大量的…     

Hamilton-Jacobi-Bellman方程的区域分解算法

对离散Hamilton-Jacobi-Bellman方程提出了一类区域分解算法,并在合理的假设下证明了该算法的单调收敛性,数值结果表明该算法的有效性与准确性．     

外区域上半线性波动方程解的整体存在性

该文研究带耗散项的线性和半线性波动方程外问题. 首先利用一个Sobolev型不等式得到了线性耗散波动方程在外区域上的整体能量衰减估计, 此结果用来证明非线性项为|u|^p (2+) 的半线性波动方程解的整体存在性. 为此, 该文主要研究N维(3≤ N≤7)外区域上球对称解的情形.     

椭圆边界上的自然积分算子及各向异性外问题的耦合算法

1.引 言为求解微分方程的外边值问题常需要引进人工边界（见[1-4]）,对人工边界外部区域作自然边界归化得到的自然积分方程即Dirichlet-Neumann映射,正是人工边界上的准确的边界条件（见[2-6]）,这是一类非局部边界条件.自然积分算子即Dirichlet-Neumann算子,     

Global smooth solutions of 3-D null-form wave equations in exterior domains with Neumann boundary conditions

The paper is devoted to investigating long time behavior of smooth small data solutions to 3-D quasilinear wave equations outside of compact convex obstacles with Neumann boundary conditions. Concretely speaking, when the surface of a 3-D compact convex obstacle is smooth and the quasilinear wave equation fulfills the null condition, we prove that the smooth small data solution exists globally provided that the Neumann boundary condition on the exterior domain is given. One of the main ingredients in the current paper is the establishment of local energy decay estimates of the solution itself. As an application of the main result, the global stability to 3-D static compressible Chaplygin gases in exterior domain is shown under the initial irrotational perturbation with small amplitude.     

Polynomial decay for a hyperbolic–parabolic coupled system

This paper analyzes the long time behavior of a linearized model for fluid-structure interaction. The space domain consists of two parts in which the evolution is governed by the heat equation and the wave equation respectively, with transmission conditions at the interface. Based on the construction of ray-like solutions by means of Geometric Optics expansions and a careful analysis of the transfer of the energy at the interface, we show the lack of uniform decay of solutions in general domains. Also, we prove the polynomial decay result for smooth solutions under a suitable Geometric Control Condition. This condition requires that all rays propagating in the wave domain reach the interface in a uniform time after, possibly, bouncing in the exterior boundary.     

Local energy decay for linear wave equations with variable coefficients

A uniform local energy decay result is derived to the linear wave equation with spatial variable coefficients. We deal with this equation in an exterior domain with a star-shaped complement. Our advantage is that we do not assume any compactness of the support on the initial data, and its proof is quite simple. This generalizes a previous famous result due to Morawetz [The decay of solutions of the exterior initial-boundary value problem for the wave equation, Comm. Pure Appl. Math. 14 (1961) 561-568]. In order to prove local energy decay, we mainly apply two types of ideas due to Ikehata-Matsuyama [L²-behaviour of solutions to the linear heat and wave equations in exterior domains, Sci. Math. Japon. 55 (2002) 33-42] and Todorova-Yordanov [Critical exponent for a nonlinear wave equation with damping, J. Differential Equations 174 (2001) 464-489].     

Global solvability for the Kirchhoff equations in exterior domains of dimension larger than three

We consider the unique global solvability of initial (boundary) value problem for the Kirchhoff equations in exterior domains or in the whole Euclidean space for dimension larger than three. The following sufficient condition is known: initial data is sufficiently small in some weighted Sobolev spaces for the whole space case; the generalized Fourier transform of the initial data is sufficiently small in some weighted Sobolev spaces for the exterior domain case. The purpose of this paper is to give sufficient conditions on the usual Sobolev norm of the initial data, by showing that the global solvability for this equation follows from a time decay estimate of the solution of the linear wave equation. Copyright © 2004 John Wiley & Sons, Ltd.     

A fast numerical method for a natural boundary integral equation for the Helmholtz equation

A Neumann boundary value problem of the Helmholtz equation in the exterior circular domain is reduced into an equivalent natural boundary integral equation. Using our trigonometric wavelets and the Galerkin method, the obtained stiffness matrix is symmetrical and circulant, which lead us to a fast numerical method based on fast Fourier transform. Furthermore, we do not need to compute the entries of the stiffness matrix. Especially, our method is also efficient when the wave number k in the Helmholtz equation is very large.     

A New Non-Overlapping Domain Decomposition Method for a 3D Laplace Exterior Problem

We propose a method for solving three-dimensional boundary value problems for Laplace’s equation in an unbounded domain. It is based on non-overlapping decomposition of the exterior domain into two subdomains so that the initial problem is reduced to two subproblems, namely, exterior and interior boundary value problems on a sphere. To solve the exterior boundary value problem, we propose a singularity isolation method. To match the solutions on the interface between the subdomains (the sphere), we introduce a special operator equation approximated by a system of linear algebraic equations. This system is solved by iterative methods in Krylov subspaces. The performance of the method is illustrated by solving model problems.