A method for solving an exterior three-dimensional boundary value problem for the Laplace equation

We develop and experimentally study the algorithms for solving three-dimensionalmixed boundary value problems for the Laplace equation in unbounded domains. These algorithms are based on the combined use of the finite elementmethod and an integral representation of the solution in a homogeneous space. The proposed approach consists in the use of the Schwarz alternating method with consecutive solution of the interior and exterior boundary value problems in the intersecting subdomains on whose adjoining boundaries the iterated interface conditions are imposed. The convergence of the iterative method is proved. The convergence rate of the iterative process is studied analytically in the case when the subdomains are spherical layers with the known exact representations of all consecutive approximations. In this model case, the influence of the algorithm parameters on the method efficiency is analyzed. The approach under study is implemented for solving a problem with a sophisticated configuration of boundaries while using a high precision finite elementmethod to solve the interior boundary value problems. The convergence rate of the iterations and the achieved accuracy of the computations are illustrated with some numerical experiments.     

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.     

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…     

On the numerical solution of singular two‐point boundary value problems: A domain decomposition homotopy perturbation approach

This paper reports a modified homotopy perturbation algorithm, called the domain decomposition homotopy perturbation method (DDHPM), for solving two‐point singular boundary value problems arising in science and engineering. The essence of the approach is to split the domain of the problem into a number of nonoverlapping subdomains. In each subdomain, a method based on a combination of HPM and integral equation formalism is implemented. The boundary condition at the right endpoint of each inner subdomain is established before deriving an iterative scheme for the components of the solution series. The accuracy and efficiency of the DDHPM are demonstrated by 4 examples (2 nonlinear and 2 linear). In comparison with the traditional HPM, the proposed domain decomposition HPM is highly accurate.     

Rational Chebyshev collocation method for the similarity solution of two dimensional stagnation point flow

In this study, we propose an efficient and accurate numerical technique that is called the rational Chebyshev collocation (RCC) method to solve the two dimensional flow of a viscous fluid in the vicinity of a stagnation point named Hiemenz flow. The Navier-Stokes equations governing the flow, are reduced to a third-order ordinary differential equation of a boundary value problem with a semi-infinite domain by using similarity transformation. The rational Chebyshev method reduces this nonlinear ordinary differential equation to a system of algebraic equations. This technique is a powerful type of the collocation methods for solving the boundary value problems over a semi-infinite interval without truncating it to a finite domain. We also present the comparison of this work with others and show that the present method is more accurate and efficient.     

Vector domain decomposition schemes for parabolic equations

A new class of domain decomposition schemes for finding approximate solutions of timedependent problems for partial differential equations is proposed and studied. A boundary value problem for a second-order parabolic equation is used as a model problem. The general approach to the construction of domain decomposition schemes is based on partition of unity. Specifically, a vector problem is set up for solving problems in individual subdomains. Stability conditions for vector regionally additive schemes of first- and second-order accuracy are obtained.     

Parallel algorithms and domain decomposition techniques for solving three-dimensional boundary value problems on quasi-structured grids

A new approach to the decomposition of a three-dimensional computational domain into subdomains matched without overlapping is proposed. It is based on direct approximation of the Poincare–Steklov equation at the interface. Parallel algorithms and techniques for solving threedimensional boundary value problems on quasi-structured grids are presented. Experimental evaluation of parallel efficiency is done by solving a model problem with quasi-structured parallelepipedal matching and non-matching grids as an example.     

Moving boundary value problems for the wave equation

We study certain boundary value problems for the one-dimensional wave equation posed in a time-dependent domain. The approach we propose is based on a general transform method for solving boundary value problems for integrable nonlinear PDE in two variables, that has been applied extensively to the study of linear parabolic and elliptic equations. Here we analyse the wave equation as a simple illustrative example to discuss the particular features of this method in the context of linear hyperbolic PDEs, which have not been studied before in this framework.     

A domain decomposition method for linear exterior boundary value problems

In this paper, we present a domain decomposition method, based on the general theory of Steklov-Poincaré operators, for a class of linear exterior boundary value problems arising in potential theory and heat conductivity. We first use a Dirichlet-to-Neumann mapping, derived from boundary integral equation methods, to transform the exterior problem into an equivalent mixed boundary value problem on a bounded domain. This domain is decomposed into a finite number of annular subregions, and the Dirichlet data on the interfaces is introduced as the unknown of the associated Steklov-Poincaré problem. This problem is solved with the Richardson method by introducing a Dirichlet-Robin-type preconditioner, which yields an iteration-by-subdomains algorithm well suited for parallel computations. The corresponding analysis for the finite element approximations and some numerical experiments are also provided.     

The Mixed Problem for the Laplace Equation in an Exterior Domain with an Arbitrary Partition of the Boundary

In this paper we propose a method for solving the mixed boundary-value problem for the Laplace equation in a connected exterior domain with an arbitrary partition of the boundary. All simple closed curves making up the boundary are divided into three sets. On the elements of the first set the Dirichlet condition is given, on the elements of the second set the third boundary condition is prescribed, and the third set, in turn, is divided into two subsets of simple closed arcs, with the Dirichlet condition prescribed on the elements of one of these subsets and the third boundary condition on the elements of the other subset. The problem is reduced to a uniquely solvable Fredholm equation of the second kind in a Banach space. The third boundary-value problem and the mixed Dirichlet--Neumann problem are particular cases of the problem under study.     

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…     

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.     

A fractional splitting algorithm for nonoverlapping domain decomposition for parabolic problem

In this article we study the convergence of the nonoverlapping domain decomposition for solving large linear system arising from semi‐discretization of two‐dimensional initial value problem with homogeneous boundary conditions and solved by implicit time stepping using first and two alternatives of second‐order FS‐methods. The interface values along the artificial boundary condition line are found using explicit forward Euler's method for the first‐order FS‐method, and for the second‐order FS‐method to use extrapolation procedure for each spatial variable individually. The solution by the nonoverlapping domain decomposition with FS‐method is applicable to problems that requires the solution on nonuniform meshes for each spatial variable, which will enable us to use different time‐stepping over different subdomains and with the possibility of extension to three‐dimensional problem. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 609–624, 2002     

On boundary value problems for the domain exterior to a thin or slender region

In this paper a method for obtaining uniformly valid asymptotic expansions of the solution of the boundary value problems in domains exterior to thin or slender regions is given. This approach combines the Tuck's method, based on the use of a suitable co-ordinates system with the method given by Handelsman and Keller yielding complete uniform asymptotic expansion of the solution for slender body problems. Our method avoids the determination of the extremities of the segment containing singularities; it is pointed out that this last problem is a pure geometrical one and independent of solving concrete boundary value problems in the given domain.     

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 investigation of a model problem for the Poisson equation with inequality constraints in a domain with a cut

A model problem is considered for the Poisson equation in a two-dimensional domain with a cut. The Dirichlet and Neumann conditions are imposed on the exterior boundary of the domain together with the nonnegativity condition for the jump across the edges of the cut. In addition, the absolute value of the gradient inside the domain must be bounded by some constant. The boundary value problem turns into a variational problem, and the unknown function must yield the minimum of the energy functional on some convex set. After discretization of the problem by the finite element method, an Uzawa-type algorithm is used to find a solution. Some examples are included of solving the discrete problem.     

A non-overlapping DDM combined with the characteristic method for optimal control problems governed by convection–diffusion equations

In this paper, we consider a non-overlapping domain decomposition method combined with the characteristic method for solving optimal control problems governed by linear convection–diffusion equations. The whole domain is divided into non-overlapping subdomains, and the global optimal control problem is decomposed into the local problems in these subdomains. The integral mean method is utilized for the diffusion term to present an explicit flux calculation on the inter-domain boundary in order to communicate the local problems on the interfaces between subdomains. The convection term is discretized along the characteristic direction. We establish the fully parallel and discrete schemes for solving these local problems. A priori error estimates in \(L^2\)-norm are derived for the state, co-state and control variables. Finally, we present numerical experiments to show the validity of the schemes and verify the derived theoretical results.     

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。     

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.     

Convergence of a numerical scheme of the type of the vortex loop method on a closed surface with an approximation to the surface shape

We consider a linear integral equation with a hypersingular integral treated in the sense of the Hadamard finite value. This equation arises when solving the Neumann boundary value problem for the Laplace equation with the use of the representation of the solution in the form of a double layer potential. We study the case in which an exterior or interior boundary value problem is solved in a domain whose boundary is a smooth closed surface and the integral equation is written out on that surface. For the numerical solution of the integral equation, the surface is approximated by spatial polygons whose vertices lie on the surface. We construct a numerical scheme for solving the integral equation on the basis of such an approximation to the surface with the use of quadrature formulas of the type of the method of discrete singularities with regularization. We prove that the numerical solutions converge to the exact solution of the hypersingular integral equation uniformly on the grid.