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…     

三维Helmholtz方程外问题的自然边界元与有限元耦合法

１．引言 设Γ０是空间闭曲面,Ω是Γ０外部的无界区域,考虑三维Ｈｅｌｍｈｏｌｔｚ 方程外Ｎｅｕｍａｎｎ问题其中 是波数,ｗ是频率,ｃ０为波在均匀介质中的传播速度,ｖ是区域Ω的边界Γ０的外法线方向,即指向由 Γ０包围的内部区域, 　　　　　　　为 Γ０上的已知函数．为了保证问题（１．１）和（１．２）的解的存在唯一性,必须附加上无穷远边界条件,即所谓的 Ｓｏｍｍｅｒｆｅｌｄ辐射条件其中ｉ是虚数单位,　　　　　． 许多数学物理问题,例如时间调和声波对不可穿透的障碍物的散射,海洋水下声波的传播,电磁波的绕射与辐…     

一类阻尼边界条件下的逆散射问题

１．引言 考虑声波在均匀非吸收介质中的传播,该声波被一个无限长的柱体所散射．在声波线性化理论中,众所周知,时间调和声波的振幅ｕ（ｘ）满足 Ｈｅｌｍｈｏｌｔｚ方程其中波数ｋ＝＞０时间调和声波的散射可以用方程（１．１）的适当的边界条件（例如Ｄｉｒｃｈｌｅｔ条件或Ｎｅｕｍａｎｎ条件）来描述．特别,ｕ｜Ｄ＝０对应于软边界（ｓｏｕｎｄ－ｓｏｆｔ）散射,而Ｄ＝ ０则是所谓的硬边界（ｓｏｕｎｄ－ｈａｒｄ）散射·这里 γ表示边界 Ｄ的外法向． 除了上述两类条件外,还有一类更能描述现实物理现象的阻尼边界条件,即物理上,…     

三维Poisson方程外问题的高阶局部人工边界条件

１引言假设Ｒ３是一分片光滑的闭曲面．是以为边界的无界区域,=Ｒ3是以为边界的有界区域,并且存在球Ｂ0＝ｘｘＲ0我们考虑下面Ｐｏｉｓｓｏｎ方程的外问题：这里f(x）,ｇ（x）是,上的已知函数,f(x)的支集是紧的,即存在一个球面=x·x=R1,使得x＝xｘＲ１,有ｆx＝０．令＝,则ｆ（ｘ）的支集包含在中,令＝xx＝,表示u在上的外法向微商．用流量为零的条件代替无限远处条件（３）,则我们得到一个新的外问题：我们将分别讨论问题（１）－（３）和（４）－（７）的数值解．由于求解区域的无界性,给数值计算带来了本质性的困难．克服此…     

椭圆外区域上Helmholtz问题的耦合法

以椭圆外区域上Helmholtz方程为例,研究一种带有椭圆人工边界的自然边界元与有限元耦合法,给出了耦合变分问题的适定性及误差分析并给出数值例子.理论分析及数值结果表明,用方法求解椭圆外问题是十分有效的.为求解具有长条型内边界外Helmholtz问题提供了一种很好的数值方法.     

利用自然边界归化求解平面弹性方程外边值问题的SCHWARZ算法

1引言平面弹性方程在水利土建等工程技术领域有着广泛应用．其中,孔边应力集中等问题,都是无界区域问题．我们可以通过各种实验手段研究上述问题．而随着计算机和有限元技术的迅猛发展,数值解法提供了一种研究上述问题的有效途径．对于有界区域上的平面弹性方程,我们可以直接利用有限元方法求解,对于其中的大规模问题可以利用区域分解和并行技术求解．但这些方法难以处理无界区域问题．虽然对于某些典型区域上的外问题（例如,圆孔外区域和＿些规则形状裂纹）可以针对具体情况利用复变函数论方法予以解决,但对于一般的无界区域问题广…     

蜕化抛物型方程的Harnack不等式

蜕化抛物型方程的Ｈａｒｎａｃｋ不等式黄清龙（兰州大学数学系，兰州７３００００）提要本文讨论一类蜕化线性抛物型方程，证明其强解具有Ｈａｒｎａｃｋ性质．关键词：抛物型微分方程；蜕化点；Ｈａｒｎａｃｋ性质ＡＭＳ（１９９１）主囹分类：３５Ｋ６５设Ｑ（６，Ｒ）...     

欧氏看涨期权定价问题的一种有效七点差分GMRES方法

用有限差分方法研究欧氏看涨期权定价问题.首先,将Black-Scholes方程通过等价代换化成一个标准的抛物型偏微分方程.其次,在求解区域构造时间精度为O（△τ^3）、空间精度为O（h^6）的差分格式,并通过Fourier分析方法证明该差分格式是无条件稳定的;边界区域选用精度较高、稳定性好的Crank-Nicolson格式,建立迭代方程.然后,用GMRES（generalized minimal residual）方法求解该方法.最后,给出一个欧氏看涨期权的数值算例,并与解析解进行比较,验证差分格式的有效性.     

关于慢增长系数非齐次线性微分方程亚纯解的级和零点

本文研究了非齐次线性微分方程ｆ（ｋ） ＋ Ｄｋ－ １ｆ（ｋ－ １） ＋ …＋ Ｄ０ｆ ＝ Ｆ （１）的复振荡问题．其中Ｄ０,…,Ｄｋ－ １是增长级小于１／２的亚纯函数,Ｆ０是有限级亚纯函数．当存在某个ＤＳ（０≤ｓ≤ｋ－ １）比其它Ｄｊ（ｊ≠ｓ）有较快增长的意义下起支配作用时,得到了微分方程（Ⅰ）的一定条件下亚纯解的级和零点的估计式．     

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．     

Discretization of distributions in the maximum domain of attraction

In this paper we discuss the discretization of distributions belonging to some max-domain of attraction. Given a random variable X its discretization is defined as the minimal integer not less than X. Our first interest is on distributions that preserve the max-domain property after discretization. Secondly, we characterize the distributions which are regarded as the discretization of the distribution in the Gumbel max-domain of attraction. Lastly the correspondence of distribution in Gumbel max-domain of attraction is investigated.     

The construction of some efficient preconditioners in the boundary element method

The discretization of first kind boundary integral equations leads in general to a dense system of linear equations, whose spectral condition number depends on the discretization used. Here we describe a general preconditioning technique based on a boundary integral operator of opposite order. The corresponding spectral equivalence inequalities are independent of the special discretization used, i.e., independent of the triangulations and of the trial functions. Since the proposed preconditioning form involves a (pseudo)inverse operator, one needs for its discretization only a stability condition for obtaining a spectrally equivalent approximation. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the discretization of a partial differential equation in the neighborhood of a periodic orbit

Sumamry This article is concerned with the comparison of the dynamic of a partial differential equation and its time discretization. We restrict our attention to the neighborhood of a hyperbolic periodic orbit. We show that the discretization possesses an invariant closed curve near the periodic orbit and that the trajectories of the semigroups defined by the partial differential equations and its approximation are close in a sense to be precised provided that different data are allowed. This answers partly an open problem posed in [4]. Examples of application to dissipative partial equations are provided.     

A thermodynamically consistent numerical method for a phase field model of solidification

A discretization is presented for the initial boundary value problem of solidification as described in the phase-field model developed by Penrose and Fife (1990) [1] and Wang et al. (1993) [2]. These are models that are completely derived from the laws of thermodynamics, and the algorithms that we propose are formulated to strictly preserve them. Hence, the discrete solutions obtained can be understood as discrete dynamical systems satisfying discrete versions of the first and second laws of thermodynamics. The proposed methods are based on a finite element discretization in space and a midpoint-type finite-difference discretization in time. By using so-called discrete gradient operators, the conservation/entropic character of the continuum model is inherited in the numerical solution, as well as its Lyapunov stability in pure solid/liquid equilibria.     

Diffusion approximation for nonparametric autoregression

A nonparametric statistical model of small diffusion type is compared with its discretization by a stochastic Euler difference scheme. It is shown that the discrete and continuous models are asymptotically equivalent in the sense of Le Cam's deficiency distance for statistical experiments, when the discretization step decreases with the noise intensity ε. Received: 12 April 1996 / Revised version: 29 October 1997     

Error estimates in elastoplasticity using a mixed method

In this paper, a mixed formulation and its discretization are introduced for elastoplasticity with linear kinematic hardening. The mixed formulation relies on the introduction of a Lagrange multiplier to resolve the non-differentiability of the plastic work function. The main focus is on the derivation of a priori and a posteriori error estimates based on general discretization spaces. The estimates are applied to several low-order finite elements. In particular, a posteriori estimates are expressed in terms of standard residual estimates. Numerical experiments are presented, confirming the applicability of the a posteriori estimates within an adaptive procedure.     

Time-discretization of nonlinear control systems with state-delay via Taylor–Lie series

The state-delay is always existent in the practical systems. Analysis of the delay phenomenon in a continuous-time domain is sophisticated. It is appropriate to obtain its corresponding discrete-time model for implementation via digital computer. In this paper, we propose a new scheme for the discretization of nonlinear systems using Taylor series expansion and the zero-order hold assumption. This scheme is applied to the sample-data representation of a nonlinear system with constant state time-delay. The mathematical expressions of the discretization scheme are presented and the effect of the time-discretization method on equilibrium properties of nonlinear control system with state time-delay is examined. The proposed scheme provides a finite-dimensional representation for nonlinear systems with state time-delay enabling existing controller design techniques to be applied to them. The performance of the proposed discretization procedure is evaluated using a nonlinear system. For this nonlinear system, various sampling rates and time-delay values are considered.     

Adaptive discretization for signal detection in statistical inverse problems

We discuss statistical tests in inverse problems when the original equation is replaced by a discretized one, i.e. a linear system of equations. Previous studies revealed that using the discretization level as regularizing procedure is possible, but its application is limited unless discretization is restricted to the singular value decomposition, see C. Marteau and P. Mathé, General regularization schemes for signal detection in inverse problems, 2013. General linear regularization may circumvent this, and we propose a regularization of the discretized equations. The discretization level may be chosen adaptively, which may save computational budget. This results in tests which are known to yield the optimal separation rate up to some constant in many cases.     

Dichotomic lattices and local discretization for Galois lattices

The present paper deals with supervised classification methods based on Galois lattices and decision trees. Such ordered structures require attributes discretization and it is known that, for decision trees, local discretization improves the classification performance compared with global discretization. While most literature on discretization for Galois lattices relies on global discretization, the presented work introduces a new local discretization algorithm for Galois lattices which hinges on a property of some specific lattices that we introduce as dichotomic lattices. Their properties, co-atomicity and \(\vee \)-complementarity are proved along with their links with decision trees. Finally, some quantitative and qualitative evaluations of the local discretization are proposed.     

L~1空间带弱奇异核的第二类Fredholm积分方程解法探究

针对带有弱奇异核的第二类Fredholm积分方程数值解法问题,介绍了两种方法.一种方法是直接用L~1空间中的离散化方法求其数值解;另一种方法是将弱奇异核通过迭代变为连续核,再用L~1空间中的离散化方法求其数值解,且通过对具体算例作图分析,从而得出直接用L~1空间中离散化方法更好.