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.     

NUMERICAL SOLUTIONS OF PARABOLIC PROBLEMS ON UNBOUNDED 3-D SPATIAL DOMAIN

In this paper,the numerical solutions of heat equation on 3-D unbounded spatial do-main are considered. n artificial boundary Γ is introduced to finite the computationaldomain.On the artificial boundary Γ,the exact boundary condition and a series of approx-imating boundary conditions are derived,which are called artificial boundary conditions.By the exact or approximating boundary condition on the artificial boundary,the originalproblem is reduced to an initial-boundary value problem on the bounded computationaldomain,which is equivalent or approximating to the original problem.The finite differencemethod and finite element method are used to solve the reduced problems on the finitecomputational domain.The numerical results demonstrate that the method given in thispaper is effective and feasible.     

A non-overlapping domain decomposition for low-frequency time-harmonic Maxwell’s equations in unbounded domains

In this paper, we are concerned with a non-overlapping domain decomposition method for solving the low-frequency time-harmonic Maxwell’s equations in unbounded domains. This method can be viewed as a coupling of finite elements and boundary elements in unbounded domains, which are decomposed into two subdomains with a spherical artificial boundary. We first introduce a discretization for the coupled variational problem by combining Nédélec edge elements of the lowest order and curvilinear elements. Then we design a D-N alternating method for solving the discrete problem. In the method, one needs only to solve the finite element problem (in a bounded domain) and calculate some boundary integrations, instead of solving a boundary integral equation. It will be shown that such iterative algorithm converges with a rate independent of the mesh size. The work of Qiya Hu was supported by Natural Science Foundation of China G10371129.     

无界区域上基于自然边界归化的一种区域分解算法

无界区域上基于自然边界归化的一种区域分解算法余德浩（中国科学院计算中心）ＡＤＯＭＡＩＮＤＥＣＯＭＰＯＳＩＴＩＯＮＭＥＴＨＯＤＢＡＳＥＤＯＮＴＨＥＮＡＴＵＲＡＬＢＯＵＮＤＡＲＹＲＥＤＵＣＴＩＯＮＯＶＥＲＵＮＢＯＵＮＤＥＤＤＯＭＡＩＮ￥ＹｕＤｅ－ｈａｏ（...     

无界区域非重叠区域分解算法的离散化及其收敛性

无界区域非重叠区域分解算法的离散化及其收敛性余德浩（中国科学院计算数学与科学工程计算研究所）ＤＩＳＣＲＥＴＩＺＡＴＩＯＮＯＦＮＯＮ－ＯＶＥＲＬＡＰＰＩＮＧＤＯＭＡＩＮＤＥＣＯＭＰＯＳＩＴＩＯＮＭＥＴＨＯＤＦＯＲＵＮＢＯＵＮＤＥＤＤＯＭＡＩＮＳＡＮＤＩ...     

空间半无界区域的非重叠区域分解算法

主要研究了空间一种半无界凹球区域上的区域分解算法.在三维空间自然边界规划的基础上,以三维Dirichlet外边值问题为例,进行的D-N交替算法.并提出了该算法与Richardson迭代法的等价性,并分析其收敛性及其收敛速度与网格参数h无关.同时给出了松弛因子的取值范围.     

FINITE ELEMENT NONLINEAR GALERKIN COUPLING METHOD FOR THE EXTERIOR STEADY NAVIER-STOKES PROBLEM

1.IntroductionNonlinearGalerkinmethodsaremultilevelschemesforthedissipativeevolutionpartialdifferentialequations.Theycorrespondtothesplittingsoftheunknownu:u=y z)wherethecomponentsareofdifferentorderofmagnitudewithrespecttoaparameterrelatedtothespati...     

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…     

无界区域上基于自然边界归化的双调和方程的一种重叠型区域分解法

1．引言双调和方程边值问题的一个力学背景是薄板弯曲问题．对于有界区域上的双调和方程，可以直接利用协调元和非协调元求解18，10］．我们考虑双调和方程Dirichlet外边值问题其中0是充分光滑闭曲线ro之外的无界区域，naro关于0的单位外法向量．引理1．且卜］．若。0EH’／‘（几），gEH‘／‘（fo），则问题（1．1）在W0z（fi）中有唯一解·这里由迹定理，可找到一个具有紧支集的函数识。。，g）E护（炉）满足可。0；g）【F。＝。0和则问题（1．1）等价于如下齐次边值问题其中f—一面‘B有紧支集．边值问题（工．2）又可转化为变分问…     

The coupling of boundary integral and finite element methods for the bidimensional exterior steady stokes problem

In this paper, we represent a new numerical method for solving the steady-state Stokes equations in an unbounded plane domain. The technique consists in coupling the boundary integral and the finite element methods. An artificial smooth boundary is introduced separating an interior inhomogeneous region from an exterior one. The solution in the exterior domain is represented by an integral equation over the artificial boundary. This integral equation is incorporated into a velocitypressure formulation for the interior region, and a finite element method is used to approximate the resulting variational problem. This is studied by means of an abstract framework, well adapted to the model problem, in which convergence results and optimal error estimates are derived. Computer results will be discussed in a forthcoming paper.     

The artificial boundary conditions for incompressible materials on an unbounded domain

Summary. In this paper we consider the numerical simulations of the incompressible materials on an unbounded domain in . A series of artificial boundary conditions at a circular artificial boundary for solving incompressible materials on an unbounded domain is given. Then the original problem is reduced to a problem on a bounded domain, which be solved numerically by a mixed finite element method. The numerical example shows that our artificial boundary conditions are very effective. ReceivedJune 7, 1995 / Revised version received August 19, 1996     

Error estimates for the finite element approximation of linear elastic equations in an unbounded domain

In this paper we present error estimates for the finite element approximation of linear elastic equations in an unbounded domain. The finite element approximation is formulated on a bounded computational domain using a nonlocal approximate artificial boundary condition or a local one. In fact there are a family of nonlocal approximate boundary conditions with increasing accuracy (and computational cost) and a family of local ones for a given artificial boundary. Our error estimates show how the errors of the finite element approximations depend on the mesh size, the terms used in the approximate artificial boundary condition, and the location of the artificial boundary. A numerical example for Navier equations outside a circle in the plane is presented. Numerical results demonstrate the performance of our error estimates.

     

一种有限元-边界元耦合分域算法

提出了一种有限元-边界元耦合分域算法．该算法将所分析问题的区域分解成有限元和边界元子域,在满足两子域界面上位移和面力协调连续的条件下,通过迭代求解得到问题的解．在迭代求解过程中,引入动态松弛系数,使收敛得以加速．该方法在两子域界面上有限单元结点和边界单元结点的位置相互独立,无需协调一致,对诸如裂纹扩展过程的模拟具有独特的优势．用所提出的耦合算法分析算例,得到的结果与有限元法、边界元法和另一种耦合算法的数值计算结果一致,验证了这种算法的正确性和可行性．     

THE ARTIFICIAL BOUNDARY CONDITION FOR EXTERIOR OSEEN EQUATION IN 2-D SPACE

A finite element method for the solution of Oseen equation in exterior domain is proposed. In this method, a circular artificial boundary is introduced to make the computational domain finite. Then, the exact relation between the normal stress and the prescribed velocity field on the artificial boundary can be obtained analytically. This relation can serve as an boundary condition for the boundary value problem defined on the finite domain bounded by the artificial boundary. Numerical experiment is presented to demonstrate the performance of the method.     

热传导方程的一类新型重叠型并行区域分解有限差分算法

0引言随着大规模科学工程计算的发展和计算精度要求的提高,区域分解和并行计算的发展越来越受到人们的重视.区域分解方法把复杂或大型的问题分解成若干重叠或非重叠子区域上的子问题,再在子区域上利用各种算法求解子问题.借助于区域分解,各子区域之间的计算可以并行,这引起了人们的研究兴趣和极大的应用前景.重叠型区域分解法的原始思想来源于Schwarz交替法.近年来建立在Schwarz交替法基础上的区域分解法在理论分析和实际应用中取得令人注目的发展,已成为一种有效的迭代方法.经典的Schwarz交替法本质上是串行的.随着并行计算的发展,出现了多种可完全并行化的Schwarz算法     

Error bounds for the finite element approximation of an incompressible material in an unbounded domain

Summary. In this paper we design high-order local artificial boundary conditions and present error bounds for the finite element approximation of an incompressible elastic material in an unbounded domain. The finite element approximation is formulated in a bounded computational domain using a nonlocal approximate artificial boundary condition or a local one. In fact there are a family of nonlocal approximate artificial boundary conditions with increasing accuracy (and computational cost) and a family of local ones for a given artificial boundary. Our error bounds indicate how the errors of the finite element approximations depend on the mesh size, the terms used in the approximate artificial boundary condition and the location of the artificial boundary. Numerical examples of an incompressible elastic material outside a circle in the plane is presented. Numerical results demonstrate the performance of our error bounds. Received August 31, 1998 / Revised version received November 6, 2001 / Published online March 8, 2002     

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.     

Three-dimensional finite/infinite elements analysis of fluid flow in porous media

Fluid flow in naturally fractured porous media can always be regarded as an unbounded domain problem and be better solved by finite/infinite elements. In this paper, a three-dimensional two-direction mapped infinite element is generated and combined with conventional finite elements and one direction infinite element to simulate poroelasticity. Therefore, the entire semi-infinite domain can be included in the numerical analysis. Both single- and dual-porosity porous media are considered. For the purpose of validation, we compare the results of finite/infinite elements with those of finite elements under two extreme boundary conditions. The comparison indicated that mapped infinite element is an appropriate approach to model fluid flow in porous media and provides an intermediate solution.     

Dirichlet-Neumann alternating algorithm for an exterior anisotropic quasilinear elliptic problem

In this paper, by the Kirchhoff transformation, a Dirichlet-Neumann (D-N) alternating algorithm which is a non-overlapping domain decomposition method based on natural boundary reduction is discussed for solving exterior anisotropic quasilinear problems with circular artificial boundary. By the principle of the natural boundary reduction, we obtain natural integral equation for the anisotropic quasilinear problems on circular artificial boundaries and construct the algorithm and analyze its convergence. Moreover, the convergence rate is obtained in detail for a typical domain. Finally, some numerical examples are presented to illustrate the feasibility of the method.     

Error analysis and preconditioning for an enhanced DtN-FE algorithm for exterior scattering problems

In this paper we present an error analysis for a high-order accurate combined Dirichlet-to-Neumann (DtN) map/finite element (FE) algorithm for solving two-dimensional exterior scattering problems. We advocate the use of an exact DtN (or Steklov–Poincaré) map at an artificial boundary exterior to the scatterer to truncate the unbounded computational region. The advantage of using an exact DtN map is that it provides a transparent condition which does not reflect scattered waves unphysically. Our algorithm allows for the specification of quite general artificial boundaries which are perturbations of a circle. To compute the DtN map on such a geometry we utilize a boundary perturbation method based upon recent theoretical work concerning the analyticity of the DtN map. We also present some preliminary work concerning the preconditioning of the resulting system of linear equations, including numerical experiments.