The boundary integral method for the steady rotating Navier–Stokes equations in exterior domain (I): the existence of solution

In this paper, we apply the boundary integral method to the steady rotating Navier–Stokes equations in exterior domain. Introducing some open ball which decomposes the exterior domain into a finite domain and a infinite domain, we obtain a coupled problem by the steady rotating Navier–Stokes equations in finite domain and a boundary integral equation without using the artificial boundary condition. For the coupled problem, we show the existence of solution in a convex set.     

The boundary integral method for the linearized rotating Navier-Stokes equations in exterior domain

In this paper, we apply the boundary integral method to the linearized rotating Navier-Stokes equations in exterior domain. Introducing some open ball which decomposes the exterior domain into a finite domain and an infinite domain, we obtain a coupled problem by the linearized rotating Navier-Stokes equations in finite domain and a boundary integral equation without using the artificial boundary condition. For the coupled problem, we show the existence and uniqueness of solution. Finally, we study the finite element approximation for the coupled problem and obtain the error estimate between the solution of the coupled problem and its approximation solution.     

半圆域内的二维线性椭圆偏微分方程

本文研究了半圆域内的二维线性椭圆偏微分方程.利用Fokas提出的求解凸多边形区域内的线性椭圆偏微分方程的变换方法,我们改进了这个方法来研究半圆域内Laplace方程,修改Helmholtz方程和Helmholtz方程的解,并且导出了这些方程解的积分表达式,讨论了Helmholtz方程的广义Dirichlet到Neumann映射.     

Forced oscillatory result of hyperbolic equations with continuous distributed deviating arguments

In this paper, we consider hyperbolic equations with continuous distributed deviating arguments, and sufficient conditions are presented for every solution of some boundary value problems to be oscillatory in a cylindrical domain. Our approach is to reduce multi-dimensional problems to one-dimensional problems by using some integral means of solution.     

Forced oscillation of certain neutral hyperbolic equations with continuous distributed deviating arguments

In this paper, we consider certain hyperbolic equations with continuous distributed deviating arguments, and sufficient conditions are presented for every solution of some boundary value problems to be oscillatory in a cylindrical domain. Our approach is to reduce multi-dimensional problems to one-dimensional problems by using some integral means of solutions.     

求解二维多区域声波传播问题的一种边界元方法

针对多区域中声波的传播问题,其中每个散射区域的介质是相同的,将散射区域内的声波用一种单双层混合位势的形式来表示,再应用Green定理表示出外部介质区域中的声波,并形成相应的边界积分方程.如果区域个数为M时,传统的边界元方法最终将形成2M个边界积分方程并对应2M个未知函数,而本文的边界元方法最终只形成M个边界积分方程以及对应M个未知函数,从而使得求解的方程和未知数的个数都减少了一倍.最后,通过对数值算例的求解,验证了该方法的可行性及精确性.     

STOKES COUPLING METHOD FOR THE EXTERIOR FLOW PART Ⅱ:WELL-POSEDNESS ANALYSIS

ＳＴＯＫＥＳＣＯＵＰＬＩＮＧ　ＭＥＴＨＯＤＦＯＲＴＨＥＥＸＴＥＲＩＯＲＦＬＯＷ　ＰＡＲＴＩＩ：ＷＥＬＬ－ＰＯＳＥＤＮＥＳＳ　ＡＮＡＬＹＳＩＳ￥ＬｉＫａｉｔａｉ（李开泰）ＢｅＹｉｎｎｉａｎ（何银年）（Ｒｏｓｅ４ｒｃｈＣｅｎｔｅｒｆｏｒＡｐｐｌｉｅｄＭａ...     

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…     

Oseen coupling method for the exterior flow. Part II: Well‐posedness analysis

In this paper, we recall the Oseen coupling method for solving the exterior unsteady Navier–Stokes equations with the non‐homogeneous boundary conditions. Moreover, we derive the coupling variational formulation of the Oseen coupling problem by using of the integral representations of the solution of the Oseen equations at an infinity domain. Finally, we provide some properties of the integral operators over the artificial boundary and the well‐posedness of the coupling variational formulation. Copyright © 2004 John Wiley & Sons, Ltd.     

A System of Plane Elasticity Canonical Integral Equations and Its Application

In this paper, we obtain a new system of canonical integral equations for the plane elasticity problem over an exterior circular domain, and give its numerical solution. Coupling with the classical finite element method, it can be used for solving general plane elasticity exterior boundary value problems. This system of highly singular equations is also an exact boundary condition on the artificial boundary. It can be approximated by a series of nonsingular integral boundary conditions.     

两连域保角映射成环域的方法

本文论述一种把型区域保角映射成环域的方法.其要点是将问题转化为Dirichlet问题,并证明该映像函数之实部应满足本文所示的边界条件,进而依据两连域上定义的调和函数的单值特性确定环域的内半径.映像函数的虚部可由Cauchy-Riemann条件得到,由此产生的积分常数仅影响映像点的幅角,并可由一一对应的映像来确定.不失其一般性,本方法可将由矩形拼成的复杂两连域保角映射成环域.笔者还对本方法作了电算,证明本方法可靠、经济、结果附有表格.     

Estimate of the modulus of continuity of a cauchy-type integral in a domain and on its boundary

We estimate the modulus of continuity of a Cauchy-type integral in a closed domain and its limit values on the boundary in the case where the boundary of the domain is an arbitrary closed rectifiable Jordan curve.     

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

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

A Nyström method for a class of Fredholm integral equations of the third kind on unbounded domains

The author proposes a numerical procedure in order to approximate the solution of a class of Fredholm integral equations of the third kind on unbounded domains. The given equation is transformed in a Fredholm integral equation of the second kind. Hence, according to the integration interval, the equation is regularized by means of a suitable one-to-one map or is transformed in a system of two Fredholm integral equations that are subsequently regularized. In both cases a Nyström method is applied, the convergence and the stability of which are proved in spaces of weighted continuous functions. Error estimates and numerical tests are also included.     

Finite Element Approximation of Conformal Mappings to Unbounded Jordan Domains

In two-dimensional Euclidean space, a simply connected unbounded domain whose boundary is a Jordan curve is called an unbounded Jordan domain. In this article, we discuss a piecewise linear finite element approximation of conformal mappings from the unit disk of the plane to unbounded Jordan domains. Some convergence results and error analysis are presented. Numerical examples are also given.     

The convergence of a direct BEM for the plane mixed boundary value problem of the Laplacian

Summary In this paper the convergence analysis of a direct boundary elecment method for the mixed boundary value problem for Laplace equation in a smooth plane domain is given. The method under consideration is based on the collocation solution by constant elements of the corresponding system of boundary integral equations. We prove the convergence of this method, provide asymptotic error estimates for the BEM-solution and give some numerical examples.     

Parallel slits map of bounded multiply connected regions

In this paper we present a boundary integral equation method for the numerical conformal mapping of bounded multiply connected region onto a parallel slit region. The method is based on some uniquely solvable boundary integral equations with adjoint classical, adjoint generalized and modified Neumann kernels. These boundary integral equations are constructed from a boundary relationship satisfied by a function analytic on a multiply connected region. Some numerical examples are presented to illustrate the efficiency of the presented method.     

One-dimensional Singular Integral Equations

A class of singular integral equations having solution in closed form is considered. This class contains the characteristic equation and other equations associated with it. Equations are solved by their reduction to some Riemann boundary value problems in generalized Hölder spaces H_φ (Γ), where Γ is a closed rectifiable Jordan curve for which the Plemelj–Privalov theorem holds.     

On the combination of some semi‐discretization methods and boundary integral equations for the numerical solution of initial boundary value problems

We consider initial boundary value problems for the homogeneous differential equation of hyperbolic or parabolic type in the unbounded two‐ or three‐dimensional spatial domain with the homogeneous initial conditions and with Dirichlet or Neumann boundary condition. The numerical solution is realized in two steps. At first using the Laguerre transformation or Rothe's method with respect to the time variable the non‐stationary problem is reduced to the sequence of boundary value problems for the non‐homogeneous Helmholtz equation. Further we construct the special integral representation for solutions and obtain the sequence of boundary integral equations (without volume integrals). For the full‐discretization of integral equations we propose some projection methods.     

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.