A Spectral Element/Laguerre Coupled Method to the Elliptic Helmholtz Problem on the Half Line

A Legendre spectral element/Laguerre coupled method is proposed to numerically solve the elliptic Helmholtz problem on the half line. Rigorous analysis is carried out to establish the convergence of the method. Several numerical examples are provided to confirm the theoretical results. The advantage of this method is demonstrated by a numerical comparison with the pure Laguerre method.     

THE GLOBAL ARTIFICIAL BOUNDARY CONDITIONS FOR NUMERICAL SIMULATIONS OF THE 3D FLOW AROUND A SUBMERGED BODY

We consider the numerical approximations of the three-dimensional steady potential flow around a body moving in a liquid of finite constant depth at constant speed and distance below a free surface in a channel. One vertical side is introduced as the up-stream artificial boundary and two vertical sides are introduced as the downstream arti-ficial boundaries. On the artificial boundaries, a sequence of high-order global artificial boundary conditions are given. Then the original problem is reduced to a problem defined on a finite computational domain, which is equivalent to a variational problem. After solving the variational problem by the finite element method, we obtain the numerical approximation of the original problem. The numerical examples show that the artificial boundary conditions given in this paper are very effective.     

A NONOVERLAPPING DOMAIN DECOMPOSITION METHOD FOR EXTERIOR 3-D PROBLEM

1. IntroductionIn recent y6ars, the elliptic boUndaly value problems ill unbounded domains have dlawnmore and more attention. TO solve an equation in an unbounded domain numerically, a basicidea is to licit the computation to a bounded domain by introducing an artWial boundary.Based on this idea, many numerical methods, such as the coupling of BEM and FEM, the FEMwith boundary conditions at atilicial boundary) the coupled finite-~ie elemellt ndhodthe DDM(domain decomposition method)(cf.,…     

ARTIFICIAL BOUNDARY METHOD FOR BURGERS＇ EQUATION USING NONLINEAR BOUNDARY CONDITIONS

This paper discusses the numerical solution of Burgers＇ equation on unbounded domains. Two artificial boundaries are introduced and boundary conditions are obtained on the artificial boundaries, which are in nonlinear forms. Then the original problem is reduced to an equivalent problem on a bounded domain. Finite difference method is applied to the reduced problem, and some numerical examples are given to show the effectiveness of the new approach.     

A POSTERIORI ESTIMATOR OF NONCONFORMING FINITE ELEMENT METHOD FOR FOURTH ORDER ELLIPTIC PERTURBATION PROBLEMS

In this paper, we consider the nonconforming finite element approximations of fourth order elliptic perturbation problems in two dimensions. We present an a posteriori error estimator under certain conditions, and give an h-version adaptive algorithm based on the error estimation. The local behavior of the estimator is analyzed as well. This estimator works for several nonconforming methods, such as the modified Morley method and the modified Zienkiewicz method, and under some assumptions, it is an optimal one. Numerical examples are reported, with a linear stationary Cahn-HiUiard-type equation as a model problem.     

GL method for solving equations in math-physics and engineering

In this paper, we propose a GL method for solving the ordinary and the partial differential equation in mathematical physics and chemics and engineering. These equations govern the acustic, heat, electromagnetic, elastic, plastic, flow, and quantum etc. macro and micro wave field in time domain and frequency domain. The space domain of the differential equation is infinite domain which includes a finite inhomogeneous domain. The inhomogeneous domain is divided into finite sub domains. We present the solution of the differential equation as an explicit recursive sum of the integrals in the inhomogeneous sub domains. Actualy, we propose an explicit representation of the inhomogeneous parameter nonlinear inversion. The analytical solution of the equation in the infinite homogeneous domain is called as an initial global field. The global field is updated by local scattering field successively subdomaln by subdomain. Once all subdomains are scattered and the updating process is finished in all the sub domains, the solution of the equation is obtained. We call our method as Global and Local field method, in short , GL method. It is different from FEM method, the GL method directly assemble inverse matrix and gets solution. There is no big matrix equation needs to solve in the GL method. There is no needed artificial boundary and no absorption boundary condition for infinite domain in the GL method. We proved several theorems on relationships between the field solution and Green＇s function that is the theoretical base of our GL method. The numerical discretization of the GL method is presented. We proved that the numerical solution of the GL method convergence to the exact solution when the size of the sub domain is going to zero. The error estimation of the GL method for solving wave equation is presented. The simulations show that the GL method is accurate, fast, and stable for solving elliptic, parabolic, and hyperbolic equations. The GL method has advantages and wide applications in the 3D electromagnetic （EM）     

ARTIFICIAL BOUNDARY METHOD FOR BURGERS'''' EQUATION USING NONLINEAR BOUNDARY CONDITIONS

This paper discusses the numerical solution of Burgers' equation on unbounded do-mains. Two artificial boundaries are introduced and boundary conditions are obtained onthe artificial boundaries, which are in nonlinear forms. Then the original problem is re-duced to an equivalent problem on a bounded domain. Finite difference method is appliedto the reduced problem, and some numerical examples are given to show the effectivenessof the new approach.     

A REMAPPING METHOD BASED ON MULTI-POINT FLUX CORNER TRANSPORT UPWIND ADVECTION ALGORITHM

A local remapping algorithm for scalar function on quadrilateral meshes is described. The remapper from a distorted grid to a rezoned grid is usually regarded as a conservative interpolation problem. The present paper introduces a pseudo time to transform the interpolation into an initial value problem on a moving grid, and construct a moving mesh method to solve it. The new feature of the algorithm is the introduction of multi- point information on each edge, which leads to the numerical flux consistent with grid node motion. During the procedure of deriving scheme, we illustrate a framework about how the algorithms on a rectangular mesh are easily generated to those on a moving mesh. The basic ideas include： （i） introducing coordinate transformation, which maps the irregular domain in physical space to a perfectly regular computational domain, and （ii） deriving finite volume methods in the physical domain, which can be viewed as a discretization of the transformed equation. The resulting scheme is second-order accurate, conservative and monotonicity preserving. Numerical examples are carried out to show the good performance of ore＂ schemes.     

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.     

NATURAL BOUNDARY ELEMENT METHOD FOR THREE DIMENSIONAL EXTERIOR HARMONIC PROBLEM WITH AN INNER PROLATE SPHEROID BOUNDARY

In this paper, we study natural boundary reduction for Laplace equation with Dirichletor Neumann boundary condition in a three-dimensional unbounded domain, which is theoutside domain of a prolate spheroid. We express the Poisson integral formula and naturalintegral operator in a series form explicitly. Thus the original problem is reduced to aboundary integral equation on a prolate spheroid. The variational formula for the reducedproblem and its well-posedness are discussed. Boundary element approximation for thevariational problem and its error estimates, which have relation to the mesh size andthe terms after the series is truncated, are also presented. Two numerical examples arepresented to demonstrate the effectiveness and error estimates of this method.     

一类各向异性外问题的重叠型区域分解算法

本文以椭圆外调和问题的自然边界归化为基础，提出了求解各向异性常系数椭圆方程的一种重叠型区域分解算法，并分析了算法的收敛性及收敛速度．理论分析及数值实验表明，该方法对于求解各向异性外问题非常有效．     

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…     

椭圆外区域上Helmholtz问题的自然边界元法

本文研究椭圆外区域上Helmholtz方程边值问题的自然边界元法.利用自然边界归化原理,获得该问题的Poisson积分公式及自然积分方程,给出了自然积分方程的数值方法.由于计算的需要,我们详细地讨论了Mathieu函数的计算方法(当0相似文献   

An iterative method based on equation decomposition for the fourth‐order singular perturbation problem

In this article, we propose an iterative method based on the equation decomposition technique ( 1 ) for the numerical solution of a singular perturbation problem of fourth‐order elliptic equation. At each step of the given method, we only need to solve a boundary value problem of second‐order elliptic equation and a second‐order singular perturbation problem. We prove that our approximate solution converges to the exact solution when the domain is a disc. Our numerical examples show the efficiency and accuracy of our method. Our iterative method works very well for singular perturbation problems, that is, the case of 0 < ε ? 1, and the convergence rate is very fast. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

解单障碍问题的非重叠区域分解法

1．引言在实际中的许多物理问题、工程问题以及各类经济平衡问题都可用变分不等式来描述．本文考虑这类问题的数值解．众所周知，区域分解法的思想可朔源到19世纪70年代提出的Schwarz交替法，但直到本世纪中期才用于数值计算．而真正获得发展还是在近十几年．由于并行机与并行算法的发展，使得Schwarz算法的优良并行性能得以开发利用，从而使得这种区域分解新技术不仅应用于偏微分方程数值解，而且广泛应用于其它各类科学与工程计算问题．近几年来，重叠型区域分解已经被成功地应用于求解椭圆型变分不等式，早期的结果见［6］．我们还可从…     

THE BOUNDARY VALUE PROBLEMS OF ELLIPTICSYSTEM (E_1) ON THE UNBOUNDED DOMAIN

1 IntroductionIn [1]. Hua, Lin and Wu developed a fullction tlieory of (A, k) bianalytic functions byinvestigatiug the second order systems of equations on the plalle. Long after a decade, it comesto know that biaualytic functions are related essentially and closely to plane elasticity. Upon80s. Gilbert and Li.[2] firstly brouglit sucli a relation to llglit. ln fact, tl1eir observation showstl1at tl1e canoliical fOrlll of tl1e elliptic systelll (E1) ili [1], wliicl1 is satisfied by (A, 1)-…     

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（...     

凹角区域双曲外问题的精确人工边界条件

研究一类凹角区域双曲型外问题的数值方法.先用Newmark方法对时间进行离散化,在每个时间步求解一个椭圆外问题.然后引入人工边界,并获得精确的人工边界条件.给出半离散化问题的变分问题,证明了变分问题的适定性,并给出了误差估计.最后给出数值例子,以示该方法的可行性与有效性.     

THE OVERLAPPING DOMAIN DECOMPOSITION METHODFOR HARMONIC EQUATION OVER EXTERIOR THREE-DIMENSIONAL DOMAIN

1.IntroductionManyscientificandengineeringproblemscallbereducedtoexteriorboundaryvalueproblemsofpartialdifferentialequations.Althoughthenumericalmethodstosolveboundaryvalueproblems,suchasthefiniteelementmethodandthefinitedifferencemethod,areveryeffectiveonboundeddomains,yetweoftenfinditdifficulttousethemtodealwithunboundedproblems.Theboundaryreductionisaforcefulmeanstosolvesomeproblemsoverunboundeddomains.Amongmanyboundaryreductions,thenaturalboundaryreductionfoundedbyK.FengandD.H.Yuhassomed…