曲边有限元和自然边界元求解外边值问题的耦合法

1　引　　言边界元与有限元耦合法在科学和工程计算中有着独特的作用 .由于区域的无限性给人们常用的有限元方法带来困难 ,边界元方法又难以独立处理非线性和非均质的问题以及具有不规则边界的区域上的问题 ,而两者相结合却可以克服各自的缺点 ,故边界元与有限元耦合法在处理一般区域问题特别是无界区域问题时便得到科学与工程界的青睐 ,获得了比较广泛的应用 .自然边界元方法并不引入新的变量 ,属于直接边界元方法[2 ] [8] .它保持能量不变和原边值问题的许多有用性质 ,例如双线性型的对称性和强制性 ,从而自然积分方程的解的存在唯一性及…     

KdV方程的特征线混合间断有限元方法

考虑KdV方程的两种特征线性混合间断有限元方法,一种方法建立在标准特征线修正方法的基础上,另一种方法是带有对流项修正的特征线修正方法.利用具有实际物理意义的特征线追踪技巧处理时间导数项和对流项,采用混合间断有限元方法处理三阶导数项,分别证明了两种方法有限元解的唯一性、稳定性和误差估计.     

任意区域的网格自动生成和任意高阶有限元方法

本文首先简要介绍非拟合网格有限元方法求解复杂区域上椭圆问题的发展现状.然后结合最近本文作者发展的非拟合网格有限元方法,针对二阶椭圆方程提出一种任意光滑区域上的任意高阶协调有限元方法.本文在带悬点的Cartesian网格上自动生成诱导网格,在诱导网格上构造协调的高阶有限元空间,采用Nitsche技术处理Dirichlet边界条件,并证明方法的适定性和hp先验误差估计.数值算例验证了本文的理论结果.     

无界区域上基于自然边界归化的一种重叠型区域分解法及其离散化

1．引言由于科学技术的迅猛发展，人们遇到许多大规模科学和工程计算问题．随着并行计算机的出现和应用，并行技术越来越得到人们的重视和研究．区域分解法成为并行计算和处理这类问题的主要方法之一．但是，对于无界区域上的椭圆边值问题，因进行区域分解后至少有一个区域仍为无界区域，故仅应用通常的区域分解算法求解是不够的．由于边界归化是处理无界区域问题的有效手段，通常采用边界元和有限元耦合的方法求解此类问题IZ，6。8。121．或片什适当的人工边界并在此边界上加近似边界条件，再在有限区域应用有限元方法求解【人习．近年来…     

分层解法——有限元的一个新算法

§.1 基本思想 有限元的出现使椭圆型方程数值解发生了巨大的变化。有限元法比差分法优越,主要表现在便于处理任意区域和任意边值。然而对于规则区域,如矩形区域,有限元就不见得比差分法优越,特别是快速福氏方法问世以后,在规则区域上,福氏方法比有限元优越     

非协调区域分解的Lagrangian乘子法

§1.引言 近年来随着并行计算机的迅速发展,求解椭圆型方程的区域分解法愈来愈引起人们的兴趣和重视.但是,目前能够见到的有限元区域分解法几乎都要求有限元空间在跨过子区域的边界时是协调的,必然限制有限元区域分解算法的优越性. [3]提出了一种非协凋区域分解法——非协调区域分解的杂交法.采用简化杂交法处理各子区域交界处的非协调性,这种方法在子区域的内部和边界采用两套不同的变量,允许内部变量在跨过各子区域的边界时不连续.但是这种方法有它的局限性,即要求边界变量在各子区域的顶点处必须保持连续性,这对推广到三维空间的情形带来很大的困难.本文提出一种非协调区域分解的Lagrangian乘子法,引进Lagrangian乘子来处理各子区域交界处的非协调性.这种方法也在子区域内部和边界采用两套不同的变量,它不仅允许内部变量在越过各子区域边界时的非协调性,并且还允许边界变量在各子区域的顶点处可以不连续,这就弥补了[3]的不足.同时,这种算法具有[3]的优点,即在不     

多孔介质中可压缩可混溶驱动问题的特征—有限元方法

在有界区域上多孔介质中可压缩可混溶的油、水两相驱动问题是由非线性偏微分方程组的初、边值问题所决定.Douglas和Roberts曾提出其数学模型并研究了半离散化方法.本文对压力方程采用有限元和混合元两种方法.对饱和度方程采用特征—有限元方法.此方法的截断误差较标准有限元小的多,随之饱和度的计算更加精确.且可用     

多孔介质中可压缩可混溶驱动问题的特征—有限元方法

在有界区域上多孔介质中可压缩可混溶的油、水两相驱动问题是由非线性偏微分方程组的初、边值问题所决定.Douglas和Roberts曾提出其数学模型并研究了半离散化方法.本文对压力方程采用有限元和混合元两种方法.对饱和度方程采用特征—有限元方法.此方法的截断误差较标准有限元小的多,随之饱和度的计算更加精确.且可用     

轴对称Poisson方程的Trefftz有限元解法

将径向基函数应用到一类轴对称Poisson方程的数值求解中,提出了一种Trefftz有限元计算格式.非0右端项将问题的特解引入Trefftz单元域内场,致使单元刚度方程涉及区域积分.利用径向基函数对特解近似处理,可消除区域积分,从而保持Trefftz有限元法只含边界积分的优势.为获得特解,选取求解域内所有单元的节点和形心作为基本插值点,而在求解域之外构造一个虚拟边界,在其上布置一定数目的虚拟点作为额外插值点.数值算例验证了该方法的有效性和可行性.     

三维热传导型半导体问题的交替方向特征有限元方法

提出交替方向特征有限元方法,对电场位势方程采用混合元格式,对电子,空穴浓度方程采用交替方向特征有限元格式,对温度方程提出交替方向格式.应用向量积计算及先验估计理论和技巧,得到最佳的L2误差估计.     

On an Area Problem in Conformal Mapping

We consider an area problem for quadrilaterals Q. The geometry of Q is partly prescribed and partly free; the module is to be fixed, and its area is to be minimized. If Q is sufficiently simple, we can solve the problem explicitly by using a modified Schwarz-Christoffel integral. The free boundary arc is represented by an integral. The corresponding area problem for doubly connected domains can be reduced to the area problem for quadrilaterals, if the given boundary component is a regular polygon. We give results of numerical experiments.     

A segmented and weighted Adomian decomposition algorithm for boundary value problem of nonlinear groundwater equation

Based on Adomian decomposition method, a new algorithm for solving boundary value problem (BVP) of nonlinear partial differential equations on the rectangular area is proposed. The solutions obtained by the method precisely satisfy all boundary conditions, except the small pieces near the four corners of the rectangular area. A theorem on the boundary error is given. Hence, the Adomian decomposition method is more efficiently applied to BVPs for partial differential equations. Segmented and weighted analytical solutions with a high accuracy for the BVP of nonlinear groundwater equations on a rectangular area are obtained by the present algorithm. Copyright © 2013 John Wiley & Sons, Ltd.     

Generating Boundary Conditions for the Calculation of Tsunami Propagation on Nested Grids

Some boundary conditions used to numerically simulate tsunami generation and propagation are studied. Special attention is given to generating boundary conditions thatmake it possible to simulate tsunami waves with desired characteristics (amplitude, time period and, in general, waveform). Since the water flow velocity in a propagating tsunami wave is uniquely defined by its height and ocean depth, one can simulate a wave propagating from the boundary into the simulation area. This can be done by specifying the wave height and water flow velocity on the boundary. This method is used to numerically simulate the propagation of a tsunami from the source to the coast on a sequence of refined grids. In this numerical experiment the wave parameters are transferred from the larger area to the subarea via boundary conditions. This method can also generate a wave that has certain characteristics on a specified line.     

Existence and geometric structure of metrics on surfaces which extremize eigenvalues

This is an exposition of the research area around our lecture at the 60^th anniversary conference of IMPA which was held in October of 2012. It is a survey of results which have been obtained over many years concerning sharp upper bounds on the first eigenvalue of a surface, either with or without boundary, in terms or area or boundary length and the surface topology. It is mostly expository, but contains a new coarse upper bound for non-orientable surfaces with boundary. It also contains a classical reformulation of recent results in [10].     

Factorization and imbedding for general linear boundary value problems

A new direction in the theory of general linear boundary value problems is explored. The starting point is an explicit Volterra factorization of the Green's matrix (and related kernels) associated with the problem. This result leads to (1) imbedding of the boundary value problems, (2) initial value algorithms for their solution, and (3) comparison theorems relating two different boundary value problems with a common boundary condition. Extensions and connections with earlier work in this area are presented.     

Filling minimality of finslerian 2-discs

We prove that every Riemannian metric on the 2-disc such that all its geodesics are minimal is a minimal filling of its boundary (within the class of fillings homeomorphic to the disc). This improves an earlier result of the author by removing the assumption that the boundary is convex. More generally, we prove this result for Finsler metrics with area defined as the two-dimensional Holmes-Thompson volume. This implies a generalization of Pu’s isosystolic inequality to Finsler metrics, both for the Holmes-Thompson and Busemann definitions of the Finsler area.     

Weak Galerkin finite element method for valuation of American options

We introduce a weak Galerkin finite element method for the valuation of American options governed by the Black-Scholes equation. In order to implement, we need to solve the optimal exercise boundary and then introduce an artificial boundary to make the computational domain bounded. For the optimal exercise boundary, which satisfies a nonlinear Volterra integral equation, it is resolved by a higher-order collocation method based on graded meshes. With the computed optimal exercise boundary, the front-fixing technique is employed to transform the free boundary problem to a one- dimensional parabolic problem in a half infinite area. For the other spatial domain boundary, a perfectly matched layer is used to truncate the unbounded domain and carry out the computation. Finally, the resulting initial-boundary value problems are solved by weak Galerkin finite element method, and numerical examples are provided to illustrate the efficiency of the method.     

On Local Nonreflecting Boundary Conditions for Time Dependent Wave Propagation

The simulation of wave phenomena in unbounded domains generally requires an artificial boundary to truncate the unbounded exterior and limit the computation to a finite region. At the artificial boundary a boundary condition is then needed, which allows the propagating waves to exit the computational domain without spurious reflection. In 1977, Engquist and Majda proposed the first hierarchy of absorbing boundary conditions, which allows a systematic reduction of spurious reflection without moving the artificial boundary farther away from the scatterer. Their pioneering work, which initiated an entire research area, is reviewed here from a modern perspective. Recent developments such as high-order local conditions and their extension to multiple scattering are also presented. Finally, the accuracy of high-order local conditions is demonstrated through numerical experiments.     

Numerical analyses of the boundary effect of radial basis functions in 3D surface reconstruction

Surface reconstruction is very important for surface characterization and graph processing. Radial basis function has now become a popular method to reconstruct 3D surfaces from scattered data. However, it is relatively inaccurate at the boundary region. To solve this problem, a circle of new centres are added outside the domain of interest. The factors that influence the boundary behaviour are analyzed quantitatively via numerical experiments. It is demonstrated that if the new centres are properly located, the boundary problem can be effectively overcome whilst not reducing the accuracy at the interior area. A modified Graham scan technique is introduced to obtain the boundary points from a scattered point set. These boundary points are extended outside with an appropriate distance, and then uniformized to form the new auxiliary centres.      

On the volume and surface area of hyperbolic polyhedra

A truncated tetrahedron is a building block of hyperbolic 3-manifolds with totally geodesic boundary. We study the relation between the volume of a truncated tetrahedron and the area of its faces which form the boundary of manifolds.