单位分解法的最优误差分析和代数多重网格法的应用
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摘要
本文的工作分两部分。第一部分主要研究了一类基于局部多项式逼近空间的单位分解法的最优误差估计。近年来,无网格方法被大量应用到科学与工程计算中。与经典有限元方法相比较,这类方法的共同特征是不再需要网格结构,它们在处理具有复杂域的问题或区域在求解过程中变化的问题时非常有效。单位分解法是非常重要的无网格方法之一,其两大主要特点是:一方面,它允许使用支集不依赖于网格或依赖于不与问题域一致的简单网格的单位分解函数(例如Shepard函数[4]),在此意义下,PUM是一种无网格方法,这个特征免去了网格生成;另一方面,局部逼近空间可以包含非多项式函数,从而很好地局部逼近未知解。这两大特征使单位分解法得到了迅速发展。虽然有关单位分解法的文献较多,但大部分都侧重于工程应用,只有极少数的数学理论分析,主要以I.Babu(?)ka和他的合作者为代表做了很多奠基性工作。但I.Babu(?)ka等现有的关于单位分解法的插值误差估计还没有获得最优阶,本文第一部分的主要工作就是:通过构造一种特殊的局部多项式近似空间,以获得最优阶插值误差估计。为此,作者从有限元方法的误差收敛阶入手,针对一类特殊的单位分解方法(取通常的有限元基函数作单位分解)进行分析,构造了一个特殊的局部多项式逼近空间,给出了一维下高次单位分解插值格式和二维下低次单位分解插值格式,推导了相应的最优阶插值误差,并研究了一维下Galerkin解的误差估计。
第二部分主要研究了求解一类椭圆型变分不等式的修正自适应代数多重网格法及其并行化。根据离散的椭圆型变分不等方程所具有的线性互补性质,提出了一个基于积极集策略之上的修正代数多重网格解法,求解具有对称二阶椭圆算子的变分不等式的有限元离散问题。数值实验表明了该算法在一致网格和h-自适应网格上的计算有效性和健壮性。为了减少计算时间,本文还根据该修正算法内在的并行度,提出了一个并行计算格式,数值结果给出了该并行的加速比和效率。
This paper consists of two parts.In the first part,we provide optimal error estimates of a class of partition of unity method(PUM) with local polynomial approximation spaces.Recently,meshfree methods have attracted much interest in the scientific computation.In contrast to classical FEM,this new family of numerical methods shares a common feature that no mesh is needed.Those methods are designed to handle more effectively problems in complex domains,or in domains evolving with the problem solution.The partition of unity method is one of important meshfree methods.One of the important aspects of PUM is that it permits the use of partition of unity functions,whose supports may not depend on any mesh(e.g.Shepard functions,see[4]),or may depend on a simple mesh that does not conform to the geometry of the domain.In this sense,the PUM is also a meshless method and this feature allows us to avoid the use of a sophisticated mesh generator.Another important aspect of PUM is that local approximation spaces can have functions other than polynomials,which locally approximate the unknown solution well.Hence PUM made a great progress.There are many recent papers on PUM,but most of them are of engineering character,without any mathematical analysis.I.Babu(?)ka and his co-workers did much fundation work.But until now I.Babu(?)ka can not get optimal order error estimates for PUM interpolants in his papers.The goal of this part is to get optimal order error estimates for PUM interpolants by chosen of a kind of special polynomial local approximation space.For this purpose,we construct a special polynomial local approximation space according to the consistence and local approximation properties of PUM at first.Then the PUM interpolation scheme of higher degree in 1D and the PUM interpolation scheme of lower degree in 2D are given,and optimal error estimates for PUM interpolants are derived.The interpolation error estimates are used to obtain optimal order error estimates for Galerkin solution in 1D.
In the second part,a modified adaptive algebraic multigrid algorithm for a class of elliptic variational inequalities and parallization are investigated.According to the linear complementarity of discrete elliptic variational inequalities,a modified algebraic multigrid(AMG) algorithm based on an active-set strategy is presented to solve the discrete problems of variational inequalities with symmetric second-order elliptic operators.The numerical experiments show the efficiency and robustness of the proposed algorithm both on the uniform mesh and on h-adaptive mesh.To shorten computation time,this part presents a parallel scheme for the modified adaptive AMG.Numerical experiments illustrate the speedup and efficiency of the parallel scheme.
第二部分主要研究了求解一类椭圆型变分不等式的修正自适应代数多重网格法及其并行化。根据离散的椭圆型变分不等方程所具有的线性互补性质,提出了一个基于积极集策略之上的修正代数多重网格解法,求解具有对称二阶椭圆算子的变分不等式的有限元离散问题。数值实验表明了该算法在一致网格和h-自适应网格上的计算有效性和健壮性。为了减少计算时间,本文还根据该修正算法内在的并行度,提出了一个并行计算格式,数值结果给出了该并行的加速比和效率。
This paper consists of two parts.In the first part,we provide optimal error estimates of a class of partition of unity method(PUM) with local polynomial approximation spaces.Recently,meshfree methods have attracted much interest in the scientific computation.In contrast to classical FEM,this new family of numerical methods shares a common feature that no mesh is needed.Those methods are designed to handle more effectively problems in complex domains,or in domains evolving with the problem solution.The partition of unity method is one of important meshfree methods.One of the important aspects of PUM is that it permits the use of partition of unity functions,whose supports may not depend on any mesh(e.g.Shepard functions,see[4]),or may depend on a simple mesh that does not conform to the geometry of the domain.In this sense,the PUM is also a meshless method and this feature allows us to avoid the use of a sophisticated mesh generator.Another important aspect of PUM is that local approximation spaces can have functions other than polynomials,which locally approximate the unknown solution well.Hence PUM made a great progress.There are many recent papers on PUM,but most of them are of engineering character,without any mathematical analysis.I.Babu(?)ka and his co-workers did much fundation work.But until now I.Babu(?)ka can not get optimal order error estimates for PUM interpolants in his papers.The goal of this part is to get optimal order error estimates for PUM interpolants by chosen of a kind of special polynomial local approximation space.For this purpose,we construct a special polynomial local approximation space according to the consistence and local approximation properties of PUM at first.Then the PUM interpolation scheme of higher degree in 1D and the PUM interpolation scheme of lower degree in 2D are given,and optimal error estimates for PUM interpolants are derived.The interpolation error estimates are used to obtain optimal order error estimates for Galerkin solution in 1D.
In the second part,a modified adaptive algebraic multigrid algorithm for a class of elliptic variational inequalities and parallization are investigated.According to the linear complementarity of discrete elliptic variational inequalities,a modified algebraic multigrid(AMG) algorithm based on an active-set strategy is presented to solve the discrete problems of variational inequalities with symmetric second-order elliptic operators.The numerical experiments show the efficiency and robustness of the proposed algorithm both on the uniform mesh and on h-adaptive mesh.To shorten computation time,this part presents a parallel scheme for the modified adaptive AMG.Numerical experiments illustrate the speedup and efficiency of the parallel scheme.
引文
[1]I.Babuska,U.Banerjee and J.E.Osborn,On principles for the selection of shape functions for the Generalized Finite Element Method,Comput.Methods Appl.Mech.Engrg.,191(2002),5595-5629.
[2]I.Babuska,U.Banerjee and J.E.Osborn,Survey of meshless and generalized finite dement methods:a unified approach,Acta Numer.,(2003),1-125.
[3]I.Babuska and J.M.Melenk,The partition of unity finite element method,Int.J.Num.Meth.Eng.,40(1997),727-758.
[4]I.Babuska,U.Banerjee and J.E.Osborn,Generalized finite element methods-main ideas,results and perspective,Int.J.Comput.Methods,1:1(2004),67-103.
[5]C.A.M.Duarte and J.T.Oden,Hp clouds-an hp meshless method,Numer.Methods Partial Different.Eqns.,12(1996),673-705.
[6]M.Griebel and M.A.Schweitzer,A particle-partition of unity method,Part Ⅱ:Efficient cover construction and reliable integration,SIAM J.Sci.Comp.,23(2002),1655-1682.
[7]M.Griebd and M.A.Schweitzer,A particle-partition of unity method,Part Ⅲ:A multilever solver,SIAM J.Sci.Comp.,24(2002),377-409.
[8]M.Griebel and M.A.Schweitzer,A particle-partition of unity method,Part Ⅳ:Parallelization,in:M.Griebel and M.A.Schweitzer(Eds.),Meshfree Methods for Partial differential Equations,Springer,2002.
[9]M.Griebel and M.A.Schweitzer,A particle-partition of unity method,Part Ⅴ:Boundary conditions,in:S.Hildebrand,H.Karcher(Eds.),Geometric Analysis and Nonlinear Partial Differential Equations,Springer,2002.
[10]W.Han and X.Meng,Error analysis of the reproducing kernel particle method,Comput.Methods Appl.Mech.Engrg.,190(2001),6157-6181.
[11]Y.Huang and J.Xu,A conforming finite element method for overlapping and nonmatching grids,Math.Comp.,72:243(2003),1057-1066.
[12]J.M.Melenk and I.Babuska,The partition of unity finite element method:Basic theorey and applications,Comput.Methods Appl.Mech.Engrg.,139(1996),289-314.
[13]T.Strouboulis,L.Zhang,D.Wang and I.Babuska,A posteriori error estimation for generalized finite element methods,Comput.Methods Appl.Mech.Engrg.,195(2006),852-879.
[14]T.Strouboulis,K.Copps and I.Babuska,Generalized finite element method,Comput.Methods Appl.Mech.Engrg.,190(2001),4081-4193.
[15]T.Strouboulis,L.Zhang and I.Babuska,Generalized finite element method using mesh-based handbooks:application to problem in domains with many voids,Comput.Methods Appl.Mech.Engrg.,192(2003),3109-3161.
[16]T.Strouboulis,L.Zhang and I.Babuska,p-Version of the Generalized FEM using mesh-based handbooks with applications to multiscale problems,Int.J.Num.Meth.Eng.,60(2004),1639-1672.
[17]I.Babuska,G.Caloz,and J.Osborn,Special finite element methods for a class of second order elliptic problems with rough coefficients,SIAM J.Numer.Anal.,31(1994),945-981.
[18]T.Strouboulis,I.Babuska,and K.Copps,The design and analysis of the generalized finite element method,Comput.Methods Appl.Meeh.Engrg.,181(2001),43-69.
[19]B.Nayroles,G.Touzot,and P.Villon,Generalizing the finite element method:diffuse approximation and diffuse elements,Comp.Mech.,10,(1992),307-318.
[20]T.Belytschko,L.Gu,and Y.Y.Lu,Fracture and crsck growth by element-free Galerkin methods,Modeling and Simulation in Material Science and Engineering,2,(1994),519-534.
[21]T.Belytschko,L.Gu,and Y.Y.Lu,Element-free Galerkin methods,Int.J.Numer.Meth.Engr.,37,(1994),229-256.
[22]W.K.Liu,S.Jun,and Y.F.Zhang,Reproducing kernel particle methods,Int.J.Numer.Meth.Engr.,20,(1995),1081-1106.
[23]W.K.Liu,S.Li,and T.Belytschko,Reproducing kernel particle method(I) Methodology and convergence,Comp.Meth.Appl.Mech.Engr.,2,(1994),519-534.
[24]Shao fan and W.K.Liu,Moving least-square reproducing Kernel Methods.part ⅱ:Fourier analysis,computer methods in applied Mechanics and Engineering,139,(1996),159-194.
[25]Lin.W.K,Shao fen Li,and T.Betytscho,Moving least-square reproducing Kernel Methods.part Ⅰ:Methodology and Convergence,Computer Methods in Applied Mechanics and Engineering,143,(1997),113-154.
[26]C.A.Duarte and J.T.Oden,An h-p adaptive method using Clouds,Comput.Meth.Appl.Mech.Engng.,139,(1996),237-262.
[27]C.A.Duarte,A review of some meshless methods to solve partial differential equations,Tech.Rep.95-06,TICAM,University of Texas,(1995).
[28]Jun Hu,Yunqing Huang,and Weimin Xue,An optimal error estimate for an hp clouds Galerkin method,Recent Progress in Computational and Applied PDEs,Conference Proceedings for the International Conference Held in ghangjiajie in July 2001,Kluwer Academic/Plenum Publishers,New York,Boston,Dordrecht,London,Moscow,(2002),217-230.
[29]Belytscho.T,Y.Y.Lu,and L.Gu,Crack propagation by element free Galerldn methods in advanced computational methods for material modeling,p.191,1993,AMD-Vol.180/PVP-vol.268,ASME 1993.
[30]I.Babuska and J.Osborn,Can a finite element method perform arbitrarily badly?Math.Comp.,69,(2000),443-462
[31]De,S.and Bathe,K.J.,The method of finite spheres,Computational Mechanics ,25,(2000),329-345
[32]Stazi,F.L.,Budyn,E.,Chessa,J.and Belytschko,T.,An extended finite element with higher-order elements for curved cracks,Computational Mechanics,31,(2003),38-48
[33]Sukumar,N.,Moes,N.,Moran,B.and Belytschko,T.,Extended finite element method for three dimensional crack modeling,Int.J.Numer.Mech.Engrg.,48,(2000),1549-1570
[34]J.M.Melenk,Finite element methods with harmonic shape function for solving Laplace's equations,Master's Thesis.University of Maryland,1992.
[35]Huang Yun-qing,Li,Wei and Su,Fang,Optimal error estimates of the partition of unity method with local polynomial approximation spaces,J.Comput.Math.,24,(2006),no.3,365-372
[36]S.N.Atluri and T.Zhu,A new meshless local Petrov-Galerkin(MLPG) approach in computational mechanics,Comput.Mech.,22,(1998),117-127
[37]G.R.Liu and Y.T.Gu,A point interpolation method,in Proc.4th Asia-Pacific Conference on Computational Mechanics,December,Singapore 1999,1009-1014
[38]G.R.Liu,A point assembly method for stress analysis for solid,in Impact Response of Materials Structures,Shim,V.P.W.et al.,Eels.,Oxford University Press,Oxford,1999,475-480
[39]E.Onate,A finite point method in computational mechanics applications to convective transport and fluid flow,Int.J.Numer.Methods Eng.,39,(1996),3839-3866
[40]T.Liszka and J.Orkisz,The finite difference method at arbitrary irregular grids and its application in applied mechanics,Comput.Struct.,11,(1980),83-95
[41]P.S.Jensen,Finite difference techniques for variable grids,Comput.Struct.,2,(1980),17-29
[42]L.Lucy,A numerical approach to testing the fission hypothesis,Astron.J.,82,(1977),1013-1024
[43]R.A.Gingold and J.J.Monaghan,Smooth particle hydrodynamics:theory and applications to nonspherical stars,Mon.Not.R.Astron.Soc.,181,(1977),375-389
[44]W.K.Liu,J.Adee and S.Jun,Reprodudng kernel and wavelet particle methods for elastic and plastic problems,in Advanced Computational Methods for Materical Modeling,Ben.son,D.J.,Ed.,180/PVP 268 ASME,1993,175-190
[45]M.G.Armentano,Error estimates in Sobolev spaces for moving least square approximation,SIAM J.Numer.Anal.,39,(1)(2002),38-51
[46]M.G.Armentano and R.G.Duran,Error estimates for moving least square approximation,Appl.Numer.Math.,To appear.
[47]T.Belytschko,W.K.Liu and M.Singer,On adaptivity and error criteria for meshfree methods,In P.Ladeveze and J.T.Oden,editors,Advances in Adaptive Computational Methods in Mechanics,1998,217-228
[48]J.H.Bramble and S.R.Hilbert,Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation,SIAM J.Numer.Anal.,7,(1970),113-124
[49]J.H.Bramble and S.R.Hilbert,Bounds on a class of linear functionals with applications to Hermite interpolation,Numer.Math.,16,(1971),362-369
[50]S.Li and W.K.Liu,Moving least squares reproducing kernel particle method:Part Ⅱ Fourier analysis,Comput.Methods Appl.Mech.Engrg.,139,(1996),159-194
[51]S.Li and W.K.Liu,Meshfree and particle methods and their application,Applied Mechanics Review,55,(2002),1-34
[52]G.R.Liu,Meshless methods,CRC Press,Boca Raton,2002
[53]A.H.Schatz and L.B.Wahlbin,Maximum norm estimates for finite element methods,Part Ⅱ.,Math.Comp.,64,(1995),907-928
[54]李荣华,冯果忱等,偏微分方程数值解法,高等教育出版社,1979年12月.
[55]王烈衡,许学军,有限元方法的数学基础,科学出版社,2005年6月.
[56]汤怀民,胡健伟,微分方程数值方法,南开大学出版社,1990年1月.
[57]付凯新,黄云清,舒适,数值计算方法,湖南科技出版社,2002年6月.
[58]M.A.Noor,K.I.Noor,Some aspects of variational inequalities,Journal of Computational and Applied Mathematics,47,(1993),285-312
[59]R.Scholz,Numerical solution of the obstacle problem by penalty method,Computing,32,(1984),297-306
[60]G.Strehlau,L_∞-error estimate for the numerical treatment of the obstacle problem by the penalty method,Number.Punct.Anal.and Optimiz,10(1/2),(1989),185-198
[61]R.Glowinski,Y.A.Kuznetsov and T.-W.Pan,A penalty/Newton/conjugate gradient method for solution of obstacle problems,C.R.Acad.Sci.Paris,Set.1336,(2003),435-440
[62]J.L.Lions and G.Stampacchia,Variational inequality,Comm,Pure Appl.Math.,20,(1967),493-519
[63]R.Glowinski,Numerical methods for nonlinear variational problems,Springer Verig New York Inc.(1984)
[64]郑铁生,李立,许庆余,一类椭圆型变分不等式离散问题的迭代算法,应用数学和力学,第16卷,第4期,(1995),329-335
[65]许学军,沈树民,障碍问题的区域分裂法,高等学校计算数学学报,2,(1994),186-194
[66]曾金平,李董辉。周叔子,解单障碍问题的非重叠区域分解法(Ⅰ):两子域情形,湖南大学学报,第24卷,第3期,(1997),1-6
[67]S.Z.Zhou,J.P.Zeng and X.Tang,Generalized schwarz algorithm for obstacle problems,Comp.and Math.with Appl.,38,(1999),263-271
[68]R.H.W.Hoppe,Multigrid algorithms for variational inequalities,SIAM J.Numer.Anat.,24(5),(1987).
[69]K.St(u|¨)ben,Algebraic multigrid(AMG):Experience and comparisions,Appl.Math.Comput.,13,(1983),419-451
[70]曾金平,马敬堂,求解一类一维椭圆型变分不等式的瀑布型多重网格法,湖南大学学报,第28卷,第5期,(2001),1-5
[71]Y.M.Zhang,Multilevel projection algorithm for solving obstacle problems,Comp.and Math.with Appl.,41,(2001),1505-1513
[72]U.Trottenberg,C.W.Oosterlee,A.Sch(u|¨)ller,A.Brandt,P.Oswald and K.St(u|¨)ben,Multigrid,Academic Press,2001
[73]R.H.W.Hoppe and R.Kornhuber,Adaptive multilevel methods for obstacle problems,SIAM J.Numer.Anal.31,(1994),301-323
[74]R.Kornhuber,Monotone multigrid methods for elliptic variational inequalities Ⅰ.,Numer.Math.69,(1994),167-184
[75]R.Kornhuber,Monotone multigrid methods for elliptic variational inequalities Ⅱ.,Numer.Math.72,(1996),481-499
[76]A.Brandt and C.W.Cryer,Multigrid algorithms for the solution of linear complementarity problems arising from free boundary problems,SIAM J.Sci.Statist.Comput.4,(1983),655-684
[77]R.Glowinski,J.L.Lions,and R.Tremolieres,Numerical Analysis of Variational inequalities,North-Holland,Amsterdam 1981
[78]W.Hackbusch and H.D.Mittelmann,On multigrid methods for variational inequalities,Numer.Math.42,(1983),65-76
[79]R.H.W.Hoppe,Multigrid algorithms for variational inequalities,SIAM J.Numer.Anal.24,(1987),1046-1065
[80]R.H.W.Hoppe,Two-sided approximations for unilateral variational inequalities by multigrid methods,Optimization 18,(1987),867-881
[81]R.H.W.Hoppe,A multilevel iterative method for symmetric,positive definite linear complementarity problems,Appl.Math.Optimization 11,(1984),77-95
[82]李若,有限元程序开发软件包AFEPACK,http://162.105.68.168
[83]Brandt,A theoritical parallel multigrid course'92,PB93,1992
[84]Douglas C.C.,A review of numerous parallel multigrid methods,Siam News,1992,25
[85]Tumina R.,Womble D.,Analysis of the multigrid FMV cycle on large-scale parallel machine,Siam J.Sci.Comput.14,(1993),1159-1173
[86]吴宏宇,并行多重网格法及其在多相流非线性分析中的应用,大连理工大学硕士学位论文
[87]都志辉,高性能计算并行编程技术一-MPI并行程序设计,清华大学出版,2001
[88]P.L.Lions and B.Mercier,Approximation numerique des equations de Hamilton-Jacobi-Bellman,Rairo Anal.Numer.,14,(1980),369-393
[89]A.Brandt,Multi-level adaptive solutions to boundary value problems,Math.Comp.,31,(1997),333-390
[90]A.Harten and J.M.Hyman,Self-adjusting grid methods for one-dimensional hyperbolic conservation laws,J.Comput.Phys.,50,(1983),235-269
[91]李晓梅,窦勇,并行计算模型及其算法设计,数值计算与计算机应用,第3期,1995,224-232
[92]程建刚,李明瑞,黄文彬,有限元分析的并行计算方法,第17卷第4期,1995,6-12
[93]R.Glowinski,Y.A.Kuznetsov and T.W.Pan,A penalty/Newton/conjugate gradient method for solution of obstacle problems,C.R.Acad.Sci.Paris,Ser.,1336,(2003),435-440
[2]I.Babuska,U.Banerjee and J.E.Osborn,Survey of meshless and generalized finite dement methods:a unified approach,Acta Numer.,(2003),1-125.
[3]I.Babuska and J.M.Melenk,The partition of unity finite element method,Int.J.Num.Meth.Eng.,40(1997),727-758.
[4]I.Babuska,U.Banerjee and J.E.Osborn,Generalized finite element methods-main ideas,results and perspective,Int.J.Comput.Methods,1:1(2004),67-103.
[5]C.A.M.Duarte and J.T.Oden,Hp clouds-an hp meshless method,Numer.Methods Partial Different.Eqns.,12(1996),673-705.
[6]M.Griebel and M.A.Schweitzer,A particle-partition of unity method,Part Ⅱ:Efficient cover construction and reliable integration,SIAM J.Sci.Comp.,23(2002),1655-1682.
[7]M.Griebd and M.A.Schweitzer,A particle-partition of unity method,Part Ⅲ:A multilever solver,SIAM J.Sci.Comp.,24(2002),377-409.
[8]M.Griebel and M.A.Schweitzer,A particle-partition of unity method,Part Ⅳ:Parallelization,in:M.Griebel and M.A.Schweitzer(Eds.),Meshfree Methods for Partial differential Equations,Springer,2002.
[9]M.Griebel and M.A.Schweitzer,A particle-partition of unity method,Part Ⅴ:Boundary conditions,in:S.Hildebrand,H.Karcher(Eds.),Geometric Analysis and Nonlinear Partial Differential Equations,Springer,2002.
[10]W.Han and X.Meng,Error analysis of the reproducing kernel particle method,Comput.Methods Appl.Mech.Engrg.,190(2001),6157-6181.
[11]Y.Huang and J.Xu,A conforming finite element method for overlapping and nonmatching grids,Math.Comp.,72:243(2003),1057-1066.
[12]J.M.Melenk and I.Babuska,The partition of unity finite element method:Basic theorey and applications,Comput.Methods Appl.Mech.Engrg.,139(1996),289-314.
[13]T.Strouboulis,L.Zhang,D.Wang and I.Babuska,A posteriori error estimation for generalized finite element methods,Comput.Methods Appl.Mech.Engrg.,195(2006),852-879.
[14]T.Strouboulis,K.Copps and I.Babuska,Generalized finite element method,Comput.Methods Appl.Mech.Engrg.,190(2001),4081-4193.
[15]T.Strouboulis,L.Zhang and I.Babuska,Generalized finite element method using mesh-based handbooks:application to problem in domains with many voids,Comput.Methods Appl.Mech.Engrg.,192(2003),3109-3161.
[16]T.Strouboulis,L.Zhang and I.Babuska,p-Version of the Generalized FEM using mesh-based handbooks with applications to multiscale problems,Int.J.Num.Meth.Eng.,60(2004),1639-1672.
[17]I.Babuska,G.Caloz,and J.Osborn,Special finite element methods for a class of second order elliptic problems with rough coefficients,SIAM J.Numer.Anal.,31(1994),945-981.
[18]T.Strouboulis,I.Babuska,and K.Copps,The design and analysis of the generalized finite element method,Comput.Methods Appl.Meeh.Engrg.,181(2001),43-69.
[19]B.Nayroles,G.Touzot,and P.Villon,Generalizing the finite element method:diffuse approximation and diffuse elements,Comp.Mech.,10,(1992),307-318.
[20]T.Belytschko,L.Gu,and Y.Y.Lu,Fracture and crsck growth by element-free Galerkin methods,Modeling and Simulation in Material Science and Engineering,2,(1994),519-534.
[21]T.Belytschko,L.Gu,and Y.Y.Lu,Element-free Galerkin methods,Int.J.Numer.Meth.Engr.,37,(1994),229-256.
[22]W.K.Liu,S.Jun,and Y.F.Zhang,Reproducing kernel particle methods,Int.J.Numer.Meth.Engr.,20,(1995),1081-1106.
[23]W.K.Liu,S.Li,and T.Belytschko,Reproducing kernel particle method(I) Methodology and convergence,Comp.Meth.Appl.Mech.Engr.,2,(1994),519-534.
[24]Shao fan and W.K.Liu,Moving least-square reproducing Kernel Methods.part ⅱ:Fourier analysis,computer methods in applied Mechanics and Engineering,139,(1996),159-194.
[25]Lin.W.K,Shao fen Li,and T.Betytscho,Moving least-square reproducing Kernel Methods.part Ⅰ:Methodology and Convergence,Computer Methods in Applied Mechanics and Engineering,143,(1997),113-154.
[26]C.A.Duarte and J.T.Oden,An h-p adaptive method using Clouds,Comput.Meth.Appl.Mech.Engng.,139,(1996),237-262.
[27]C.A.Duarte,A review of some meshless methods to solve partial differential equations,Tech.Rep.95-06,TICAM,University of Texas,(1995).
[28]Jun Hu,Yunqing Huang,and Weimin Xue,An optimal error estimate for an hp clouds Galerkin method,Recent Progress in Computational and Applied PDEs,Conference Proceedings for the International Conference Held in ghangjiajie in July 2001,Kluwer Academic/Plenum Publishers,New York,Boston,Dordrecht,London,Moscow,(2002),217-230.
[29]Belytscho.T,Y.Y.Lu,and L.Gu,Crack propagation by element free Galerldn methods in advanced computational methods for material modeling,p.191,1993,AMD-Vol.180/PVP-vol.268,ASME 1993.
[30]I.Babuska and J.Osborn,Can a finite element method perform arbitrarily badly?Math.Comp.,69,(2000),443-462
[31]De,S.and Bathe,K.J.,The method of finite spheres,Computational Mechanics ,25,(2000),329-345
[32]Stazi,F.L.,Budyn,E.,Chessa,J.and Belytschko,T.,An extended finite element with higher-order elements for curved cracks,Computational Mechanics,31,(2003),38-48
[33]Sukumar,N.,Moes,N.,Moran,B.and Belytschko,T.,Extended finite element method for three dimensional crack modeling,Int.J.Numer.Mech.Engrg.,48,(2000),1549-1570
[34]J.M.Melenk,Finite element methods with harmonic shape function for solving Laplace's equations,Master's Thesis.University of Maryland,1992.
[35]Huang Yun-qing,Li,Wei and Su,Fang,Optimal error estimates of the partition of unity method with local polynomial approximation spaces,J.Comput.Math.,24,(2006),no.3,365-372
[36]S.N.Atluri and T.Zhu,A new meshless local Petrov-Galerkin(MLPG) approach in computational mechanics,Comput.Mech.,22,(1998),117-127
[37]G.R.Liu and Y.T.Gu,A point interpolation method,in Proc.4th Asia-Pacific Conference on Computational Mechanics,December,Singapore 1999,1009-1014
[38]G.R.Liu,A point assembly method for stress analysis for solid,in Impact Response of Materials Structures,Shim,V.P.W.et al.,Eels.,Oxford University Press,Oxford,1999,475-480
[39]E.Onate,A finite point method in computational mechanics applications to convective transport and fluid flow,Int.J.Numer.Methods Eng.,39,(1996),3839-3866
[40]T.Liszka and J.Orkisz,The finite difference method at arbitrary irregular grids and its application in applied mechanics,Comput.Struct.,11,(1980),83-95
[41]P.S.Jensen,Finite difference techniques for variable grids,Comput.Struct.,2,(1980),17-29
[42]L.Lucy,A numerical approach to testing the fission hypothesis,Astron.J.,82,(1977),1013-1024
[43]R.A.Gingold and J.J.Monaghan,Smooth particle hydrodynamics:theory and applications to nonspherical stars,Mon.Not.R.Astron.Soc.,181,(1977),375-389
[44]W.K.Liu,J.Adee and S.Jun,Reprodudng kernel and wavelet particle methods for elastic and plastic problems,in Advanced Computational Methods for Materical Modeling,Ben.son,D.J.,Ed.,180/PVP 268 ASME,1993,175-190
[45]M.G.Armentano,Error estimates in Sobolev spaces for moving least square approximation,SIAM J.Numer.Anal.,39,(1)(2002),38-51
[46]M.G.Armentano and R.G.Duran,Error estimates for moving least square approximation,Appl.Numer.Math.,To appear.
[47]T.Belytschko,W.K.Liu and M.Singer,On adaptivity and error criteria for meshfree methods,In P.Ladeveze and J.T.Oden,editors,Advances in Adaptive Computational Methods in Mechanics,1998,217-228
[48]J.H.Bramble and S.R.Hilbert,Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation,SIAM J.Numer.Anal.,7,(1970),113-124
[49]J.H.Bramble and S.R.Hilbert,Bounds on a class of linear functionals with applications to Hermite interpolation,Numer.Math.,16,(1971),362-369
[50]S.Li and W.K.Liu,Moving least squares reproducing kernel particle method:Part Ⅱ Fourier analysis,Comput.Methods Appl.Mech.Engrg.,139,(1996),159-194
[51]S.Li and W.K.Liu,Meshfree and particle methods and their application,Applied Mechanics Review,55,(2002),1-34
[52]G.R.Liu,Meshless methods,CRC Press,Boca Raton,2002
[53]A.H.Schatz and L.B.Wahlbin,Maximum norm estimates for finite element methods,Part Ⅱ.,Math.Comp.,64,(1995),907-928
[54]李荣华,冯果忱等,偏微分方程数值解法,高等教育出版社,1979年12月.
[55]王烈衡,许学军,有限元方法的数学基础,科学出版社,2005年6月.
[56]汤怀民,胡健伟,微分方程数值方法,南开大学出版社,1990年1月.
[57]付凯新,黄云清,舒适,数值计算方法,湖南科技出版社,2002年6月.
[58]M.A.Noor,K.I.Noor,Some aspects of variational inequalities,Journal of Computational and Applied Mathematics,47,(1993),285-312
[59]R.Scholz,Numerical solution of the obstacle problem by penalty method,Computing,32,(1984),297-306
[60]G.Strehlau,L_∞-error estimate for the numerical treatment of the obstacle problem by the penalty method,Number.Punct.Anal.and Optimiz,10(1/2),(1989),185-198
[61]R.Glowinski,Y.A.Kuznetsov and T.-W.Pan,A penalty/Newton/conjugate gradient method for solution of obstacle problems,C.R.Acad.Sci.Paris,Set.1336,(2003),435-440
[62]J.L.Lions and G.Stampacchia,Variational inequality,Comm,Pure Appl.Math.,20,(1967),493-519
[63]R.Glowinski,Numerical methods for nonlinear variational problems,Springer Verig New York Inc.(1984)
[64]郑铁生,李立,许庆余,一类椭圆型变分不等式离散问题的迭代算法,应用数学和力学,第16卷,第4期,(1995),329-335
[65]许学军,沈树民,障碍问题的区域分裂法,高等学校计算数学学报,2,(1994),186-194
[66]曾金平,李董辉。周叔子,解单障碍问题的非重叠区域分解法(Ⅰ):两子域情形,湖南大学学报,第24卷,第3期,(1997),1-6
[67]S.Z.Zhou,J.P.Zeng and X.Tang,Generalized schwarz algorithm for obstacle problems,Comp.and Math.with Appl.,38,(1999),263-271
[68]R.H.W.Hoppe,Multigrid algorithms for variational inequalities,SIAM J.Numer.Anat.,24(5),(1987).
[69]K.St(u|¨)ben,Algebraic multigrid(AMG):Experience and comparisions,Appl.Math.Comput.,13,(1983),419-451
[70]曾金平,马敬堂,求解一类一维椭圆型变分不等式的瀑布型多重网格法,湖南大学学报,第28卷,第5期,(2001),1-5
[71]Y.M.Zhang,Multilevel projection algorithm for solving obstacle problems,Comp.and Math.with Appl.,41,(2001),1505-1513
[72]U.Trottenberg,C.W.Oosterlee,A.Sch(u|¨)ller,A.Brandt,P.Oswald and K.St(u|¨)ben,Multigrid,Academic Press,2001
[73]R.H.W.Hoppe and R.Kornhuber,Adaptive multilevel methods for obstacle problems,SIAM J.Numer.Anal.31,(1994),301-323
[74]R.Kornhuber,Monotone multigrid methods for elliptic variational inequalities Ⅰ.,Numer.Math.69,(1994),167-184
[75]R.Kornhuber,Monotone multigrid methods for elliptic variational inequalities Ⅱ.,Numer.Math.72,(1996),481-499
[76]A.Brandt and C.W.Cryer,Multigrid algorithms for the solution of linear complementarity problems arising from free boundary problems,SIAM J.Sci.Statist.Comput.4,(1983),655-684
[77]R.Glowinski,J.L.Lions,and R.Tremolieres,Numerical Analysis of Variational inequalities,North-Holland,Amsterdam 1981
[78]W.Hackbusch and H.D.Mittelmann,On multigrid methods for variational inequalities,Numer.Math.42,(1983),65-76
[79]R.H.W.Hoppe,Multigrid algorithms for variational inequalities,SIAM J.Numer.Anal.24,(1987),1046-1065
[80]R.H.W.Hoppe,Two-sided approximations for unilateral variational inequalities by multigrid methods,Optimization 18,(1987),867-881
[81]R.H.W.Hoppe,A multilevel iterative method for symmetric,positive definite linear complementarity problems,Appl.Math.Optimization 11,(1984),77-95
[82]李若,有限元程序开发软件包AFEPACK,http://162.105.68.168
[83]Brandt,A theoritical parallel multigrid course'92,PB93,1992
[84]Douglas C.C.,A review of numerous parallel multigrid methods,Siam News,1992,25
[85]Tumina R.,Womble D.,Analysis of the multigrid FMV cycle on large-scale parallel machine,Siam J.Sci.Comput.14,(1993),1159-1173
[86]吴宏宇,并行多重网格法及其在多相流非线性分析中的应用,大连理工大学硕士学位论文
[87]都志辉,高性能计算并行编程技术一-MPI并行程序设计,清华大学出版,2001
[88]P.L.Lions and B.Mercier,Approximation numerique des equations de Hamilton-Jacobi-Bellman,Rairo Anal.Numer.,14,(1980),369-393
[89]A.Brandt,Multi-level adaptive solutions to boundary value problems,Math.Comp.,31,(1997),333-390
[90]A.Harten and J.M.Hyman,Self-adjusting grid methods for one-dimensional hyperbolic conservation laws,J.Comput.Phys.,50,(1983),235-269
[91]李晓梅,窦勇,并行计算模型及其算法设计,数值计算与计算机应用,第3期,1995,224-232
[92]程建刚,李明瑞,黄文彬,有限元分析的并行计算方法,第17卷第4期,1995,6-12
[93]R.Glowinski,Y.A.Kuznetsov and T.W.Pan,A penalty/Newton/conjugate gradient method for solution of obstacle problems,C.R.Acad.Sci.Paris,Ser.,1336,(2003),435-440