关于Schwarz交替法的拟边界松弛方法

§1.松弛方法 我们讨论二阶自共轭椭圆型方程的Dirichlet问题.设Ω?R~2为一多边形区域. a(u,v)=(f,v),v∈H_0~1(Ω),f∈H~(-1)(Ω), u∈H_0~1(Ω)是定义在其上的边值问题的变分形式,这里取齐次边界条件仅为叙述问题方便.双线性型a(·,·)满足:     

关于并行迭代区域分解算法的松弛因子

1引言考虑二阶椭圆型Dirichlet边值问题的弱形式,求u∈H_0~1(Ω)使得a(u,v)=(f,v),(?) v∈H_0~1(Ω),(1)其中Ω是平面多角形区域,f∈L~2(Ω),(f,v)=∫_Ωfvdx,a(u,v)=∫_Ω(sum from i,j=1 to 2 a_(ij)(?)u/(?)x_i(?)等 a_0uv)dx,其中[a_(ij)]在Ω上对称一致正定,a_(ij)在Ω上分片连续有界,a_0≥0．由Lax-Milgram引理,问题(1)在H_0~1(Ω)中有唯一解．     

拟线性椭圆型问题有限元近似解的迭代加速分析

一、问题的提出 我们考察二阶拟线性椭圆型第一边值问题: -?(α(x,u)?u)=f(x,u),在Ω内, u(x)=0,在?Ω上,其中Ω是R~n(n=2,3)中有界开区域,?Ω是Ω的光滑边界。若u(x),α(x,u(x))和f(x,u(x))有足够正规性,则问题(1)的等价弱形式方程是:对于u∈H_0~1(Ω), (α(x,u)?u,?v)=(f(x,u),v),?v∈H_0~1(Ω)。 (2)这里假设α(x,u)在Ω×R中为正的且有界,内积     

椭圆型方程的并行迭代区域分裂法——两个子区域情形

§1.问题的分析 设Ω?R~2是一有界开区域,是定义在Ω上的椭圆算子,其中对X∈Ω,[a_(i·j)(X)]_i,j=1,2对称且一致正定;a_(ij)(X)分片连续且上,下有界,a(X)≥0.我们求解如下问题: Lu=f,在Ω中, u=0,在?Ω上, (1.1)其中f∈H~(-1)(Ω),u∈H_0~1(Ω).这里取齐次Dirichlet边界条件,仅仅是为了叙述问题的方便.(1.1)的变分形式是     

三维光滑区域上Navier-Stokes方程的有限元方法

§1.引言设Ω是R~3中的有界区域,且属于C~2.v是正常数.已知:当f∈(L~2(Ω))~3.且||f||0充分小时,Navier—Stokes方程的解存在且唯一.另外,u∈(H~2(Ω)∩H_1~0(Ω))~3,P∈H~1(Ω)\R. 最近,Bernardi讨论三维多面体区域Ω上Stokes方程的有限元解法.有限元空间由分片多项式(关于u为分片特殊三次多项式,关于p为分片常数)构成,进行误差估计时.要求u∈(H~2(Ω)∩H_0~1(Ω))~3,P∈H~1(Ω)\R.当Ω为二维区域上的凸多角形时,Stokes     

简单分歧数值解的L_p估计

§1.问题的提出 考虑单参数二阶椭圆拟线性微分方程:λ∈R,Ω?R~N(N=1,2)是多角形凸域(要求?Ω是Lipschitz连续的)或光滑域.[a_(ij)]∈C~1满足正定条件.f(x,y)∈C~2f(x,0)≡0,f_y(x,θ)≥0,但f_y(x,0)?0,?x∈Ω.记||·||_(j,p,Ω),p≥1,j=0,1,2为通常的W~(j,p)(Ω)范.H_0~1?W_0~(1,2),(·,·)为H_0~j中通     

简单分歧数值解的L_p估计

§1.问题的提出 考虑单参数二阶椭圆拟线性微分方程:λ∈R,Ω?R~N(N=1,2)是多角形凸域(要求?Ω是Lipschitz连续的)或光滑域.[a_(ij)]∈C~1满足正定条件.f(x,y)∈C~2f(x,0)≡0,f_y(x,θ)≥0,但f_y(x,0)?0,?x∈Ω.记||·||_(j,p,Ω),p≥1,j=0,1,2为通常的W~(j,p)(Ω)范.H_0~1?W_0~(1,2),(·,·)为H_0~j中通     

一类非线性双曲-抛物耦合问题的A.D.I.Galerkin方法

1　引　言考虑三维非线性双曲 -抛物耦合初边值问题 :utt- . (a1 (X,t,u) u) +b1 (X,t,u,v) . u　　　　 +α1 e. v =f(X,t,u,v) ,X∈Ω,t∈ J.vt-a2 Δv +b2 (X,t,u,v) . v　　　　 +α2 e. ut=g(X,t,u,v) ,X∈Ω,t∈ J.u(X,t) =v(X,t) =0 , X∈ Ω ,t∈ J.u(X,0 ) =u0 (X) ,ut(X,0 ) =ut0 (X) ,v(X,0 ) =v0 (X) ,X∈Ω.(1 .1 )其中 ,X=(x1 ,x2 ,x3) ,Ω=(c1 ,d1 )× (c2 ,d2 )× (c3,d3)为 R3中矩形区域 ,边界 Ω . J=[0 ,T] ,T>0为一正常数 .b1 ,b2 ,f,g均为已知光滑函数 (其中 b1 ,b2 为向量函数 ) ,且关于 u,v满足 L…     

一类带临界指数增长的椭圆型方程组两个正解的存在性

该文主要讨论带临界指数的椭圆型方程组{-Δu + a(x)u =2α/α+βuα-1vβ + f(x),x ∈Ω,-Δv+b(x)v=2β/α+βuαvβ-1+ g(x),x ∈ Ω,(*)u ＞ 0,v ＞ 0,x ∈Ω,u=v=0,x ∈(a)Ω解的存在性,其中Ω是RN中一个光滑有界区域,N=3,4,a≥2,β≥2...     

Jacobian-determinant equation in the critical Besov spaces

Let Ω be a bounded domain in R~n with smooth boundary. Here we consider the following Jacobian-determinant equation det u(x)=f(x),x∈Ω;u(x)=x,x∈?Ω where f is a function on Ω with min_Ω f = δ 0 and Ωf(x)dx = |Ω|. We prove that if f ∈B_(p1)~(np)(Ω) for some p∈(n,∞), then there exists a solution u ∈ B_(p1)~(np+1)(Ω)C~1(Ω) to this equation. On the other hand, we give a simple example such that u ∈ C_0~1(R~2, R~2) while detu does not lie in B_(p1)~(2p)(R~2) for any p∞.     

用CROUZEIX－RAVIART元解非自共轭椭圆型问题的重叠型区域分解算法

用ＣＲＯＵＺＥＩＸ－ＲＡＶＩＡＲＴ元解非自共轭椭圆型问题的重叠型区域分解算法顾金生（北京航空航天大学动力系）胡显承（清华大学应用数学系）ＯＶＥＲＬＡＰＰＩＮＧＤＯＭＡＩＮＤＥＣＯＭＰＯＳＩＴＩＯＮＭＥＴＨＯＤＦＯＲＮＯＮＳＥＬＦＡＤＪＯＩＮＴＥＬＬＩ...     

一个双调和方程的Schwarz交替法

设Ω为IR~2平面上的有界区域,其边界(?)Ω适当光滑,考虑四阶调和方程: △~2表示双调和算子,f∈L~2(Ω).(1.1)式的物理模型为简支板的平衡方程,问题解的存在唯一性在[5]中已有证明.     

基于半平面上自然边界归化的无界区域上的Schwarz交替法及其离散化

In this paper,we discuss a Schwarz alternating method for a kind of unboundeddomains, which can be decomposed into a bounded domain and a half-planar domain. Finite Element Method and Natural Boudary Reduction are used alternatively. The uniform geometric convergence of both continuous and discrete problems is proved. The theoretical results as well as the numerical examples show thatthe convergence rate of this discrete Schwarz iteration is independent of the finiteelement mesh size, but dependent on the overlapping degree of subdomains.     

On Schwarz Alternating Methods for the Subsonic Full Potential Equation

The Schwarz alternating method can be used to solve elliptic boundary value problems on domains which consist of two or more overlapping subdomains. The solution is approximated by an infinite sequence of functions which results from solving a sequence of elliptic boundary value problems in each of the subdomains. The full potential equation is derived from the Navier–Stokes equations assuming the fluid is compressible, inviscid, irrotational and isentropic. It is being used by the aircraft industry to model flow over an airfoil or even an entire aircraft. This paper shows that the additive and multiplicative versions of the Schwarz alternating method, when applied to the full potential equation in three dimensions, converge to the true solution geometrically. The assumptions are that the initial guess and the true solution are everywhere subsonic. We use the convergence proof by Tai and Xu and modify it for certain closed convex subsets.     

On the geometric convergence of optimized Schwarz methods with applications to elliptic problems

The Schwarz method can be used for the iterative solution of elliptic boundary value problems on a large domain Ω. One subdivides Ω into smaller, more manageable, subdomains and solves the differential equation in these subdomains using appropriate boundary conditions. Optimized Schwarz Methods use Robin conditions on the artificial interfaces for information exchange at each iteration, and for which one can optimize the Robin parameters. While the convergence theory of classical Schwarz methods (with Dirichlet conditions on the artificial interface) is well understood, the overlapping Optimized Schwarz Methods still lack a complete theory. In this paper, an abstract Hilbert space version of the Optimized Schwarz Method (OSM) is presented, together with an analysis of conditions for its geometric convergence. It is also shown that if the overlap is relatively uniform, these convergence conditions are met for Optimized Schwarz Methods for two-dimensional elliptic problems, for any positive Robin parameter. In the discrete setting, we obtain that the convergence factor ρ(h) varies like a polylogarithm of h. Numerical experiments show that the methods work well and that the convergence factor does not appear to depend on h.     

波动方程外区域问题基于自然边界归化的区域分解算法(英文)

本文以二维波动方程为例 ,研究基于自然边界归化的一种区域分解算法 .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格式 ,对每一时间步长求解一椭圆型外问题 ;然后引入两条人工边界 ,提出了 Schwarz交替算法 ,给出了算法的收敛性 ,并对圆外区域研究了压缩因子     

Splitting a Concave Domain to Convex Subdomains

1.IntroductionTurninglargescaleproblemtosmallscalesubproblemsandregularizingirregularproblemaretwomainsubjectsofdomaindecomposition.Inregularization,regularizingirregularregionisoffirstimportance.Irregularityoftenmeansconcavity,forexample,L-shaped,T--shapedandC-shapeddomainsareirregulardomains.Inthispaperswewillstudydomaindecompositionmethodforellipticproblemsdefinedonirregularregion.SchwarzalternatingInethodisthebasisofalmostalldomaindecompositionmethoddeveloped.Othermethodsarevariationsof…     

各向异性外问题的Schwarz 交替法及其收敛性和误差估计

本文对于无界区域各向异性常系数椭圆型偏微分方程研究了一种基于自然边界归化的Schwarz交替法.利用极值原理证明了在连续情形最大模意义下的几何迭代收敛性,通过选取适当的共焦椭圆边界利用Fourier分析获得了不依赖各向异性程度的最优的迭代收缩因子,还在离散情形最大模意义下证明了几何收敛性,而且进一步得到了误差估计,最后,数值结果证实了迭代收缩因子和误差估计的正确性,表明了该方法在无界Ⅸ域上求解各向异性椭圆型偏微分方程的优越性.     

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.