几类基于几何和分析信息的代数多重网格法及其应用
|
摘要
多重网格法是求解偏微分方程大规模离散化方程的最为有效的方法，粗略地讲，它可分为几何多重网格法和代数多重网格(AMG)法。这里，我们将结合几何和代数两种途径来研究多重网格法，并称之为所谓的基于几何和分析的代数多重网格法，这是目前国际上代数多重网格法研究领域中新发展起来的方法。
本文分为两部分。第一部分，结合几何和代数多重网格法的特点，我们为两类典型的复杂有限元方程组，设计了具有很强的Robust性和高效性的代数多重网格法。第一类是R~d，d=2，3中的高次Lagrange有限元方程组，其系数矩阵的具有较强稠密性：另一类是所谓的Criss-Cross网格下的线性有限元方程组的约化线性子系统，其相应的几何粗空间一般不具有嵌套性。通过对问题和有限元空间作深入、细致的分析，发现了许多重要的代数特征，形成了一些新的关于求解复杂有限元方程组的网格粗化技术和提升算子构造的代数方法，从本质上克服了粗网格层自由度难以控制等通常代数多重网格法的缺陷，该类AMG法还具有预处理(Setup)时间少、Robust性好和运算效率高等特性。进一步，通过引入新的证明方法，即利用所谓的Xu-Zikatanov恒等式等，我们从理论上严格证明了新算法的最优收敛(下降)率，数值试验验证了理论的正确性。另外，通过引入两套代数矩阵，对高次Lagrange有限元方程组，我们设计并分析了相应的基于AMG法的预条件共轭梯度法。这些主要的算法设计思想和理论分析方法，具有相当的普适性。
第二部分，我们针对两种应用问题，讨论和分析相应的代数多重网格法。第一种是晶格材料的离散模型。我们首先设计了一种基于AMG法的块预条件共轭梯度法，并就方形晶格模型，利用其近似连续模型，从理论上严格证明了其关于参数α是一致收敛性。接着又构造了对更广泛的晶格模型具有高效性和Robust性的AMG法和相应的APCG法，数值试验表明我们的算法对许多晶格模型，关于其规模和重要参数α是一致收敛的。第二种应用问题来源于辐射流体力学方程组，我们讨论其中的二维三温能量方程离散系统的代数多重网格法。我们针对二维三温能量方程的特殊性，建立了一种半粗化的代数多重网格法(SAMG)和以该SAMG为预条件子的Krylov子空间迭代法，并将其嵌入到能量方程与流体力学方程耦合后得到的应用程序中，通过与经典预条件GEMRES(m)和ORTHOMIN(m)迭代法作对比数值实验，表明我们的AMG方法具有高效性和很好的Robust性。
Multigrid methods are by far the most efficient methods for solving large scale algebraic systems arising from discretizations of partial differential equations. Roughly speaking, these methods can be developed though two approaches: geometric approach and algebraic approach. The purpose of this work is to develop multigrid methods by combining both geometric and algebraic approaches, which may be known as algebraic multigrid(AMG) methods based on geometric and analytic information.
This dissertation consists of two parts. In the first part, by combining advantages of both geometric and algebraic multigrid methods, some robust multigrid methods are constructed for two kinds of finite element equations, one is the high order Lagrangian finite element equation for which AMG is often not very efficient; another is a condensed finite element system on criss-cross grids where the corresponding coarse spaces can not be made nested easily. By an effective use of geometric information and analytic properties of underlying differential equations and finite element spaces, many algebraic features can be exploited for developing new coarsening techniques and new interpolation operators. The new method overcomes the difficulty for properly controlling the degrees of freedom of the coarse spaces in the usual AMG methods. Numerical results show that our algebraic multigrid algorithm is substantially better than many usual algebraic multigrid algorithms. Furthermore, by using a new theoretical approach, namely the Xu-Zikatanov identity, a rigorous convergence analysis of our algebraic multigrid method is given. In addition, based on an algebraic multigrid method of linear finite elements, a robust preconditioned conjugate gradient method is presented and analyzed for the discrete systems of high order Lagrangian finite elements. These new ideas of hybrid multigrid and theoretical analysis provided in this work can be extended to more general cases.
In the second part, algebraic multigrid methods are applied to solve two kinds of discrete systems arising from practical applications. First a block preconditioned conjugate gradient method (BPCG) and a class of algebraic multigrid methods are developed and studied for some discrete mathematical models for lattice block materials. Numerical experiments show that the new AMG methods converge uniformly with respect to the size of problem and also to some crucial parameters. Such a uniform convergence of the BPCG algorithm is further theoretically justified properly by
本文分为两部分。第一部分，结合几何和代数多重网格法的特点，我们为两类典型的复杂有限元方程组，设计了具有很强的Robust性和高效性的代数多重网格法。第一类是R~d，d=2，3中的高次Lagrange有限元方程组，其系数矩阵的具有较强稠密性：另一类是所谓的Criss-Cross网格下的线性有限元方程组的约化线性子系统，其相应的几何粗空间一般不具有嵌套性。通过对问题和有限元空间作深入、细致的分析，发现了许多重要的代数特征，形成了一些新的关于求解复杂有限元方程组的网格粗化技术和提升算子构造的代数方法，从本质上克服了粗网格层自由度难以控制等通常代数多重网格法的缺陷，该类AMG法还具有预处理(Setup)时间少、Robust性好和运算效率高等特性。进一步，通过引入新的证明方法，即利用所谓的Xu-Zikatanov恒等式等，我们从理论上严格证明了新算法的最优收敛(下降)率，数值试验验证了理论的正确性。另外，通过引入两套代数矩阵，对高次Lagrange有限元方程组，我们设计并分析了相应的基于AMG法的预条件共轭梯度法。这些主要的算法设计思想和理论分析方法，具有相当的普适性。
第二部分，我们针对两种应用问题，讨论和分析相应的代数多重网格法。第一种是晶格材料的离散模型。我们首先设计了一种基于AMG法的块预条件共轭梯度法，并就方形晶格模型，利用其近似连续模型，从理论上严格证明了其关于参数α是一致收敛性。接着又构造了对更广泛的晶格模型具有高效性和Robust性的AMG法和相应的APCG法，数值试验表明我们的算法对许多晶格模型，关于其规模和重要参数α是一致收敛的。第二种应用问题来源于辐射流体力学方程组，我们讨论其中的二维三温能量方程离散系统的代数多重网格法。我们针对二维三温能量方程的特殊性，建立了一种半粗化的代数多重网格法(SAMG)和以该SAMG为预条件子的Krylov子空间迭代法，并将其嵌入到能量方程与流体力学方程耦合后得到的应用程序中，通过与经典预条件GEMRES(m)和ORTHOMIN(m)迭代法作对比数值实验，表明我们的AMG方法具有高效性和很好的Robust性。
Multigrid methods are by far the most efficient methods for solving large scale algebraic systems arising from discretizations of partial differential equations. Roughly speaking, these methods can be developed though two approaches: geometric approach and algebraic approach. The purpose of this work is to develop multigrid methods by combining both geometric and algebraic approaches, which may be known as algebraic multigrid(AMG) methods based on geometric and analytic information.
This dissertation consists of two parts. In the first part, by combining advantages of both geometric and algebraic multigrid methods, some robust multigrid methods are constructed for two kinds of finite element equations, one is the high order Lagrangian finite element equation for which AMG is often not very efficient; another is a condensed finite element system on criss-cross grids where the corresponding coarse spaces can not be made nested easily. By an effective use of geometric information and analytic properties of underlying differential equations and finite element spaces, many algebraic features can be exploited for developing new coarsening techniques and new interpolation operators. The new method overcomes the difficulty for properly controlling the degrees of freedom of the coarse spaces in the usual AMG methods. Numerical results show that our algebraic multigrid algorithm is substantially better than many usual algebraic multigrid algorithms. Furthermore, by using a new theoretical approach, namely the Xu-Zikatanov identity, a rigorous convergence analysis of our algebraic multigrid method is given. In addition, based on an algebraic multigrid method of linear finite elements, a robust preconditioned conjugate gradient method is presented and analyzed for the discrete systems of high order Lagrangian finite elements. These new ideas of hybrid multigrid and theoretical analysis provided in this work can be extended to more general cases.
In the second part, algebraic multigrid methods are applied to solve two kinds of discrete systems arising from practical applications. First a block preconditioned conjugate gradient method (BPCG) and a class of algebraic multigrid methods are developed and studied for some discrete mathematical models for lattice block materials. Numerical experiments show that the new AMG methods converge uniformly with respect to the size of problem and also to some crucial parameters. Such a uniform convergence of the BPCG algorithm is further theoretically justified properly by
引文
[1] R.A. Adams, Sobolev Spaces, Academic Press, New York, 1975.
[2] P. G. Ciarlet, The Finite Element for Elliptic Problems, North-Holland, Amsterdam, New York, Oxford, 1978.
[3] Chen Chuanmiao, Huang Yunqing, High Accuracy Theory of FEM(in Chinese), Changsha: Hunan Science and Technology Press, 1995.
[4] Huang Yunqing, Shu Shi and Yu Haiyuan, Superconvergence and asymptotic expansions for linear finite element approximations on criss-cross mesh. Science in China Ser. A Mathematics 2004 Vol. 47, Supp. 136-145.
[5] 林群，严宁宁，高效有限元构造与分析，河北大学出版社，1996．
[6] 应隆安，有限元方法讲义，北京大学出版社，1988．
[7] 朱起定，林群，有限元超收敛理论，湖南科学技术出版社，1989．
[8] R. E. Alcouffe, A. Brandt, J.E. Dendy, and J. W. Painter, The Multi-Grid Methods for the Diffusion Equation With Strongly Discontinuous Coefficients, SIAM J. Sci. Stat. Comput. 2 (1981), pp. 430-454.
[9] G. P. Astrakhantsev, An Iterative Method of Solving Elliptic Net Problems, USSR Comp. Math. Math. Phys. 11 (2), 171-182, 1971.
[10] N.S. Bakhvalov, Convergence of a Relaxation Method With Natural Constraints on the Elliptic Operator, USSR, Comp. Math. Math. Phys. 6, 101-135, 1966.
[11] R. Bank, T. Dupont and H. Yserentant, The Hierarchical Basis Multigrid Method. Numer. Math. 52, 427-458, 1988.
[12] R. Bank, and J. Xu, An Algorithm for Coarsing Unstructured Meshes, Numer. Math., 73, No. 1, pp. 1-36, 1996.
[13] F. Bornemann and P. Deuflhard, The Cascadic Multigrid Method for Elliptic Problems, Numer. Math. 75, 135-152, 1996.
[14] D. Braess, The Contraction Number of a Multigrid Method for Solving the Poission Equation. Numer. Math. 37, 387-404, 1981.
[15] D. Braess, W. Hackbusch, A new Convergence Proof for the Multigrid Method Including the V-Cycle. SIAM J. Numer. Anal. 20, 967-975, 1983.
[16] J. H. Bramble, Multigrid Methods, Pitman Research Notes in Mathematical Series 294, Longman, Harlow, 1993.
[17] J. H. Bramble, J. E. Pasciak, J. Wang, and J. Xu, Convergence Estimates for Multigrid Algorithms Without Regularity Assumptions, Math. Comp., 57 (1991), pp. 23-45.
[18] J.H. Bramble, J. E. Pasciak, and J. Xu, Parallel Multilevel Preconditions, Math.Comp. 55 (1990), 1-22.
[19] J.H. Bramble, J. E. Pasciak, and J. Xu, The Analysis of Algorithms With Nonnested Spaces or Noninhereted Quadratic Forms, Math. Comp., 56 (1991),pp. 1-34.
[20] A.Brandt, Multi-level Adaptive Technique (MLAT) for Fast Numerical Solution to Boundary Value Problems, Proceedings of the 3rd International Conference on Numerical Methods in Fluid Mechanics (Paris,1972), Lecture Notes in Physics,18(eds H. Cabannes and R. Temam), Springer-Verlag, Berlin and New York,1973, pp. 82-89.
[21] A.Brandt, Multi-level Adaptive Techniques (MLAT), I. The Multigrid Method,Research Rep. RC 6026, IBM T.J. Watson Research Center, Yorktown Heights,NY, 1976.
[22] A.Brandt, Multi-Level Adaptive Solutions to boundary-Value Problems, Math.Comput. 31, 333-390, 1977.
[23] A.Brandt, Multigrid Techniques: 1984 Guide With Applications to Fluid Dynamics, GMD-Studien Nr. 85, Gesellschaft fur Mathematik und Datenverar-beitung, St.Augustin, 1984.
[24] A. Brandt, Rigorous Quantitative Analysis of Multigrid: I. Constant Coefficients Two Level Cycle With L_2 Norm, SIAM J. Numer. Anal. 31, 1695-1730, 1994.
[25] A.Brandt, B. Diskin, Multigrid Solvers for Non-Aligned Sonic Flows. SIAM J.Sci. Comput. 21, 473-501, 1999.
[26] A.Brandt, and I. Yavneh, On Multigrid Solution of High-Reynolds Incompressible Entering Flows. J. Comp. Phys. 101, 151-164, 1992.
[27] A.Brandt, and I. Yavneh, Accelerated Multigrid Convergence and High Recircu-lating Flows. SIAM J. Sci. Comput. 14, 607-626, 1993.
[28] S. Brenner, Convergence of the Multigrid V-Cycle Algorithm for Second-Order Boundary Value Problems Without Full Elliptice Regularity, Math. Comp. 71(2002), pp. 507-525 (electroic).
[29] W.L. Briggs, A Multigrid Tutorial, Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, 1987 (W.L. Briggs, V.E. Henson, and S.F.McCormick, A Multigrid Tutorial, 2nd Edition, SIAM, Philadelphia, 2000).
[30] C. Buldwin, P.N. Brown, R.D. Falgout, J. Jones, and F. Gramam, Iterative Linear Solvers in a 2D Radiation-Hydrodynamics Code : Methods and Performance,J. Comput. Phys., 154 (1999), pp. 1-40.
[31] R.H. Chan, T.F. Chan, and W.L. Wan, Multigrid for Differential-Convolution Problems Arising From Image Processing, Technical Report CAM 97-20, Department of Mathematics, UCLA, 1997.
[32] T.F. Chan, H.B. Keller, Arc-Length Continuation and Multi-Grid Techniques for Nonlinear Elliptic Eigenvalue Problems, SIAM J. Sci. Comput. 3. 173-194,1982.
[33] T.F. Chan, S. Go, and L. Zikatanov, Multigrid Methods for Elliptic Problems on Unstructured Grids, VKI Lecture Series 1997-02, Van Karman Institute, Rhode St. G. Belgium, 1997.
[34] J.E. Dendy, Jr., M..P. Ida, and J. M. Rutledge. A Semi-Coarsening Multigrid Algorithm for SIMD Machines, SIAM J. Sci. Stat. Comput., 13: 1460-1469,1992.
[35] J.E. Dendy, S.F. McCormick, J.W. Ruge, T.F. Russel, and S. Schaffer, Multi-grid Methods for Three-Dimensional Petroleum Reservoir Simulation, Paper SPE 18409, Tenth Symposium on Reservoir Simulation, Houston, Texas, 1989.
[36] E. Dick, J. Linden, A Multigrid Method for Steady Incompressible Navier-Stokes Equations Based on Flux Difference Splitting, Int. J. Num. Meth. Fluids 14,1311-1323, 1992.
[37] E. Dick, K. Riemslagh, Multi-Staging of Jacobi Relaxation to Improve Smoothing Properties of Multigrid Methods for Steady Euler equations, II. J. Comp. Appl.Math. 59, 339-348, 1995.
[38] E. Dick, and J. Steelant, Coupled Solution of the Steady Compressible Navier-Stokes Equations and the k — e Turbulence Equations With a Multigrid Method,Appl. Numer. 23, 49-61, 1997.
[39] E. Dick, K. Riemslagh, and J. Vierendeels, Multigrid Methods VI, Proceedings of the 6th European Multigrid Conference, Lectured Notes in Computational Science and Engineering 14, Springer, Berlin, 2000.
[40] Y. R. Efendiev, T. Y. Hou, and Z.-H. Wu. Convergence of a Nonconforming Multiscale Finite Element Method. SIAM J. Numer. Anal., 37 (3): 888-910 (electronic), 2000.
[41] B. Engquist and E. Luo, Convergence of a Multigrid Method for Elliptic Equations With Highly Oscillatory Coefficients, SIAM J. Numer. Anal., 34: 2254-2273, 1997.
[42] R.P. Fedorenko, A Relaxation Method for Solving Elliptic Difference Equations,USSR Comp. Math. Math. Phys., 1 (5), 1092-1096, 1962.
[43] R.P.Fedorenko, On the Speed of Convergence of an Iterative Process, USSR Comp. Math. Math. Phys., 4 (3), 227-235, 1964. MR 31#6386.
[44] H. Foerster, K. Stuben, and U. Trottenberg, Non-Standard Multigrid Techniques Using Checkered Relaxation and Intermediate Grids. Elliptic Problem Solvers (ed. M. H. Schultz), 285-300, Academic Press, New York, 1981.
[45] P. O. Frederickson, and O. A. McBryan, Parallel Superconvergent Multigrid, Multigrid Methods: Theory, Applications and Supercomputing (ed. S. Mc-Cormick), 195-210, Marcel Dekker, New York, 1988.
[46] P. O. Frederickson, and O. A. McBryan, Recent Developments for the PSMG Multiscale Method. Multigrid Methods Ⅲ, Proceedings of the 3rd International Conference on Multigrid Methods (eds W. Hackbusch and U. Trottenberg). 21-40, Birkhauser, Basle, 1991.
[47] A. Greenbaum, A Multigrid Method for Multiprocessors, Appl. Math. Comp. 19, 23-45, 1986.
[48] H. Guillard, and P. Vanek, An aggregation Multigrid Solver for Convection-Diffusion Problems on Unstructured Meshes, Rep. 130, Center for Computational Mathematics, University of Denver, 1998.
[49] H. Guillard, and G. W. Zumbusch, Parallel Multigrid in an Adaptive PDE Solver Based on Hashing and Space-Filling Curves, Parallel Comput. 25, 827-843, 1999.
[50] W. Hackbusch, Ein Iteratives Verfahren Zur Schnellen Auflosung Elloptischer Randwertproblem, Rep. 76-12, Institute for Applied Mathematics, University of Cologne, West Germany, 1976.
[51] W. Hackbusch, On the Convergence of a multi-grid iteration applied to finite element equations. Rep. 77-8, Institute for Applied Mathematics, University of Cologne, West Germany, 1977.
[52] W. Hackbusch, On the Multigrid Method Applied to Diffrence Equations, Computing, 20 (1978), pp. 291-306.
[53] W. Hackbusch, Convergence of Multi-Grid Iterations Applied to Difference Equations. Rep. 77-8, Math., Comp. 34. 425-440, 1980.
[54] W. Hackbusch, On the Convergence of a Multi-Grid Iteration. Beitrage Numer. Math. 9, 213-239, 1981
[55] W. Hackbusch, Multigrid Methods and Applications, Vol.4 of Computational Mathematics, Springer-Verlag, Berlin,1985.
[56] W. Hackbusch, The Frequency Decomposition Multigrid Method, part Ⅰ: Appllication to Anisotropic Equations, Numer. Math., 56: 229-245, 1989.
[57] W. Hackbusch and U. Trottenberg (eds), Multigrid Methods, Lecture Notes in Mathematics 960. Springer, Berlin, 1982.
[58] W. Hackbusch and U. Trottenberg (eds), Multigrid Methods II. Lecture Notes in Mathematics 1228. Springer, Berlin, 1986.
[59] W. Hackbusch and U. Trottenberg (eds), Multigrid Methods III. Proceedings of the 3rd International Conference on Multigrid Methods, International Series on Numerical Mathematics 98, Birkhauser, Basle, 1991.
[60] W. Hackbusch and G. Wittum, Multigrid Methods V, Proceedings of the 5th European Multigrid Conference, Lecture Notes in Computational Science and Engineering 3. Springer, Berlin, 1998.
[61] P.W. Hemker, On the Order of Prolongations and Restrictions in Multigrid Procedures, J. Comp. Appl. Math. 32, 423-429, 1990.
[62] P.W. Hemker and P. Wesseling (eds.), Multigrid Methods IV, Proceedings of the 4th European Multigrid Conference. Birkhauser, Basle, 1994.
[63] H. Hoppe, and H. Miihlenbein, Parallel Adaptive Full-Multigrid Methods on Message-Based Multiprocessors, Parallel Comput. 3, 269-287, 1986.
[64] G. Horton, The Time-Parallel Multigrid Method, Comm. Appl. Num. Meth. 8.585-596, 1992.
[65] Yunqing Huang, Zhongci Shi, Tao Tang and Weimin Xue, A multilevel successive iteration method for nonlinear elliptic problems. Math. Comp. 73 (2004), 525-539.
[66] W. Joppich and S. Mijalkovie, Multigrid Methods for Process Simulation.Springer, Vienna, 1993.
[67] R. Kettler, Analysis and Comparison of Relaxation Schemes in Robust Multigrid and Preconditioned Conjugate Gradient Methods, In W. Hackbusch and U. Trottenberg, Editors, Multigrid Methods, pages 502-534, Springer-Verlag, November1981.
[68] H. Kim, J. Xu, and L. Zikatanov, Uniformly Convergent Multigrid Methods for Convection Diffusion Problems Without any Constraint on Coarse Grids, Advanced in Comp. Math., (2002). (to appear).
[69] O. Kolp, and H. Mierendorff, Efficient Multigrid Algorithms for Locally Con-strainted Parallel Systems, Appl. Math. Comp. 19, 169-200, 1986.
[70] B. Koobus, M. H. Lallemand, and A. Dervieux, Unstructured Volume-Agglomeration MG: Solution of the Poisson Equation, International Journal for Numerical Methods in fluids 18 (1994), No.1, 27-42.
[71] F.B. Lin, and F. Sotiropoulos, Strongly-Coupled Multigrid Method for 3D Incompressible Flows Using Near-Wall Turbulence Closures, Trans. ASME J. Fluids Eng. 119, 314-324, 1997.
[72] J. Linden, B. Steckel, K. Stuben, Parallel Multigrid Solution of the Navier-Stokes Equations on General 2D-Domains, Parallel Comput. 7, 461-475, 1988.
[73] J. Linden, G. Lonsdale, H. Ritzdorf, and A. Schuller, Scalability Aspects of Parallel Multigrid. Fut. Generation Comp. Systems 10, 429-439, 1994.
[74] O.A. MaBryan, P.O. frederickson, J. Linden, A. Schuller, K. Solchenbach, K.Stuben, C.A. Thole, U. Trottenberg, Multigrid Methods on Parallel Computers a Survey of Recent Developments, Impact Comput. Sci. Eng. 3, 1-75, 1991.
[75] D.J. Mavripilis, V. Venkatakrishnan, Agglomeration Multigrid for Two-Dimensional Viscous Flows, Comp. Fluids 24, 553-570, 1995.
[76] D.J. Mavripilis, V. Venkatakrishnan, A 3D Agglomeration Multigrid Solver for the Reynolds Averaged Navier-Stokes Equations on Unstructured Meshes, Int.J. Num. Meth. Fluids 23, 527-544, 1996.
[77] S.F. McCormick, Multigrid Methods for variational Problems : Further Results,SIAM J.Numer. Anal. 21 (1984), pp. 255-263.
[78] S.F. McCormick, Multigrid Methods for Variational Problems: General. Theory for the V-Cycle, SIAM J. Numer. Anal. 22(1985), pp. 634-643.
[79] S.F. McCormick, Multigrid Methods. SIAM, Philadelphia, 1988.
[80] S.F. McCormick, and J.W. Ruge, Multigrid Methods for Variational Problems:SIAM J. Numer. Anal. 19 (1982), pp. 924-929.
[81] S.F. McCormick, Multilevel Adaptive Methods for Partial Differential Equations. Frontiers in Applied Mathematics 6. SIAM Philadelphia, 1989.
[82] H. Mierendorff, Parallelization of Multigrid Methods With Local Refinements for a Class of Nonshared Memory Systems, Multigrid Methods: Theory, Applications and Supercomputing (ed. S.F. McCormick), 449-465, Marcel Dekker, New York, 1988.
[83] E. Morano, D.J. Mavripilis and V.C. Venkatakrishnan, Coarsening Strategies for Unstructured Multigrid Techniques With Application to Anisotropic Problems, 7th Copper Mountain Conference on Multigrid Methods, NASA Conference Publication 3339 (eds N.D. Melson, T.A. Manteuffel, S.F. McCormick, and C.C.Douglas), 591-606, NASA, Hampton, 1996.
[84] N.H. Naik, and J. van Rosendale, The Improved Robustness of Multigrid Elliptic Solvers Based on Multiple Semicoarsened Grids, SIAM J. Numer. Anal. 30, 215-229, 1993.
[85] R.A. Nicolaides, On Multiple Grid and Related Techniques for Solving Discrete Elliptic Systems, J. Comp. Phys. 19, 418-431, 1975.
[86] J.E. Pasciak, and J. Xu. Parallel Multilevel Preconditioners, Math. Comp., 55(1990), pp. 1-22.
[87] A.A. Reusken, A Multigrid Method Based on Incomplete Gaussian Elimination,Numer. Linear Algebra Appl. 3 (1996) , No. 5, 369-390
[88] V.V. Shaidurov, Multigrid Methods for Finite Elements, Kluwer, Dordrecht,1995.
[89] Zhong-ci Shi and Xue-jun Xu, Cascadic Multigrid Method for Elliptic Problems,East-West J.Numer.Math., 7 (1999), 199-211.
[90] Zhong-ci Shi and Xue-jun Xu, Cascadic Multigrid for Parabolic Problems, J.Comput. Math. , 18 (2000), 551-560.
[91] Zhong-Ci Shi, Xuejun Xu, A new Cascadic Multigrid, Sciences in China Series 4, Vol. 44, No. 1, pp. 21-30, 2001.
[92] R.A. Smith and A. Weiser, Semicoarsening Multigrid on a Hypercube, SIAM J.Sci. Stat. Comput., 13: 1314-1329, 1992.
[93] S.P. Spekreijse, Multigrid Solution of Second Order Discretizations of Hyperbolic Conservation laws, Math. Comp. 49, 135-155, 1987.
[94] K. Solchenbach, and U. Trottenberg, On the Multigrid Acceleration Approach in Computational Fluid Dynamics, Arbertspapiere der GMD 251, Sankt Augustin,West Germanv, 1987.
[95] C.A. Thole, and U. Trottenberg, A Short Note on Standard Multigrid Algorithms for 3D Problems, Appl. Math. Comp. 27 (2), 101-111, 1988.
[96] U. Trottenberg and A. Schuller, Multigrid, Academic Press Inc., San Diego, CA,2001. With Contributions by A.Brandt,P.Oswald and K.Stuben.
[97] P. Vanek, M. Brezina, and R. Telaur, Two-grid Method for Linear Elasticity on Unstructured Meshes, SIAM J. Sci. Comput., 21 (1999), pp. 900-923.
[98] F. Wang, J. Shen, and J. Xu, A Finite Element Multigrid Preconditioner for Chebyshev-Collocation Methods, Appl. Numer. Math., 33 (2002), pp. 471-477.
[99] T. Washio, C.W. Oosterlee, Krylov Subspace Acceleration for Nonlinear Multi-grid Schemes, Electr. Trans. Num. Anal. 6, 271-290, 1997.
[100] P. Wesseling, An Introduction to Multigrid Methods, John Wiley and Sons,Chichester, 1992.
[101] G. Wittum, Multi-Grid Methods for Stokes and Navier-Stokes Equations With Transforming Smoothers: Algorithms and Numerical results, Numer. Math. 54,543-563, 1989.
[102] J. Wu, H. Ritzdorf, C.W. Oosterlee, B. Steckel, and A. Schuller, Adaptive Parallel Multigrid Solution of 2D Incompressible Navier-Stokes Equations, Int. J. Num. Methods Fluids 24, 875-892, 1997.
[103] J. Xu, Theory of Multilevel Methods, PHD thesis, Cornell University, Ithaca, N. Y., 1989.
[104] J. Xu, Iterative Methods by Space Decomposition and Subspace Correction, SIAM Rev., 34 (1992), pp. 581-613.
[105] J. Xu, A Novel Two-Grid Method for Semilinear Equations, SIAM J. Numer. Anal, 33, (1996), 1759-1777.
[106] J. Xu, The Method of Subspace Corrections, J. Comp. Appl. Math., 128 (2001), pp. 335-362.
[107] J. Xu and A. Zhou, A Two-Grid Discretization Scheme for Eigenvalue Problems, Math. Comp., 70(2001), pp. 17-25.
[108] J. Xu and A. Zhou, Local and Parallel Finite Element Algorithms Based on Two-Grid Discretizations, Math. Comp., 231 (2001), pp. 881-909.
[109] J. Xu and A. Zhou, Locall and Parallel Finite Element Algorithms Based on Two-Grid Discretizations for Nonlinear Problems, Advances in Comp. Math., (to appear)
[110] J. Xu and L. Zikatanov, On Multigrid Methods for Generalized Finite Element Methods, in Proc. Intl. Workshop Meshfree Meth. for PDE, Vol. 26 of Springer Lect. Notes on Comp. Sci. Eng. series, 2002.
[111] I. Yavneh, Multigrid Factors for Red-Black Gauss-Seidel Relaxation Applied to a Class of Elliptic Operators, Numer. Anal. 32, 1126-1138, 1995.
[112] I. Yavneh, On Red-Black SOR Smoothing in Multigrid, SIAM J. Sci. Comput. 17, 180-192, 1996.
[113] I. Yavneh, C.H. Venner, A. Brandt, Fast Multigrid Solution of the Advection Problem With Closed Characteristics, SIAM J. Sci. Comput. 19, 111-125, 1998.
[114] H. Yserentant, On the Multi-Level Splitting of Finite Element Spaces, Numer. Math. 49, 379-412, 1986.
[115] H. Yserentant, Old and new Convergence Proofs for Multigrid Methods. Acta Numerica 2,285-326, 1993.
[116] O. Axelsson, P.S. Vassilevski, Algebraic Multilevel Preconditioning Methods Ⅰ, Num. Math. 56, 157-177, 1989.
[117] D. Braess, Towards Algebraic Multigrid for Ellipitic Problems of Sencond Order, Computing 55 (1995), 379-393
[118] A. Brandt, Algebraic Multigrid Theory: the Symmetric Case, Appl. Math.Comp. , 19, 23-56, 1986
[119] A. Brandt, General Highly Accurate Alegebraic Coarsening Schemes, Proceedings of the 9th Copper Mountain on Multigrid Methods, Copper Mountain, April 1999.
[120] A. Brandt. Multiscale Scientific Computation. Six Year Research Summary.1999.
[121] A. Brandt, Multiscale Scientific Computation: Review 2001, in Multiscale and Multiresolution Methods, Vol. 20 of Lect. Notes Comput. Sci. Eng., Springer,Berlin, 2002, pp. 3-95.
[122] A.Brandt, S.F. McCormick, Algebraic Multigrid (AMG) for Sparse Matrix Equations, in "Sparsity and Its Application", D.J. Evans (ed.), Cambridge University Press, pp. 257-284, Cambridge, 1984.
[123] A. Brandt, S.F. McCormick, and J.W. Ruge, Algebraic multigrid (AMG) for Automatic Multigrid Olution with Application to Geodetic Computations, Institute for Computational Studies, POB 1852, Fort Collins, Colorado, 1982.
[124] A.Brandt, S.F. McCormick, and J.W. Ruge, Multigrid Methods for Differential Eigenproblems, SIAM J. Sci. Stat. Comput., 4 (1983), pp. 244-260.
[125] Qianshun Chang, et al., On the Algebraic Multigrid Method, J. Comput.Physics, 125, 279-292, 1996.
[126] Qianshun Chang, Zhaohui Huang, Efficient Algebraic Multigrid Algorithm and Their Convergence, SIAM J. Sci. Comput. Vol. 24, No. 2, pp. 597-618, 2002.
[127] A.J. Cleary, R.D. Falgout, V.E. Henson, J.E. Jones, Coarse-Grid Selection for Parallel Algebraic Multigrid. Proceedings of the 5th International Symposium on Solving Irregularly Structured Problems in Parallel, Lecture Notes in Computer Science 1457, 104-115, Springer, New York, 1998.
[128] A.J. Cleary, R.D. Falgout, V.E. Henson, J.E. Jones, T.A. Manteuifei, S.F. McCormick, G.N. Miranda, and J.W. Ruge, Robustness and Scalability of Algebraic Multigrid, SIAM J.Sci. Comput. (special Issue on the Fifth Copper Mountain Conference on Iterative Methods, 1998). 21, 1886-1908, 2000.
[129] W. Dahmen, L. Eisner, Algebraic Multigrid Methods and the Schur Complement, Notes on Numerical Fluid Mechanics 23, Vieweg, Braunshweig, 1988.
[130] P.M. de Zeeuw, Matrix-Dependent Prolongations and Restrictions in a Blackbox Multigrid Solver, J. Comput. Appl. Math. 33 (1990), 1-27.
[131] J.E. Dendy, Black Box Multigrid, J. Comput. Phys., 48, 366-386, 1982.
[132] J.E. Dendy, Black Box Multigrid for Nonsymmetric Problems, Appl. Math.Comp. 13, 261-284, 1983.
[133] M. Griebel, T. Neunhoeffer, and H. Regler, Algebraic Multigrid Methods for the Solution of the Navier-Stokes Equations in Complicated Geometries,SFB-Bericht Nr. 342/01/96 A, Institutfur Informatik, Technische Universitat Munchen, Munich, 1996.
[134] R.D. Lonsdale, An Algebraic Multigrid Solver for the Navier-Stokes Equations on Unstructured Meshes, Int. J. Num. Meth. Heat Fluid Flow
[135] W.Z. Huang, Convergence of Algebraic Multigrid Methods for Symmetric Positive Definite Matrices With Weak Diagonal Dominance, Appl. Math, and Comp.,46, pp. 145-164, 1991.
[136] F. Kickinger, Algebraic Multi-Grid for Discrete Elliptic Second Order Problems,Institutsbericht 513, Institut fur Mathematik, Universitat Linz, Austria, 1997.
[137] A. Krechel, K. Stuben, Operator Dependent Interpolation in Algebraic Multi-grid, Multigrid Methods V, Proceedings of the 5th European Multigrid Conference, Lecture Notes in Computational Science and Engineering 3 (eds W.Hackbusch and G. Wittum). 189-211, Springer, Berlin, 1998.
[138] A. Krechel, K. Stuben, Parallel Algebraic Multigrid Based on subdomain Blocking. GMD-Report 71, 1999. Submitted to Parallel Comput.
[139] S.F. McCormick, and J.W. Ruge, Algebraic Multigrid Methods Applied to Problems in Computational Structural Mechanics, State-of-the-Art Surveys on Computational Mechanics, 237-270, ASME, New York, 1989.
[140] R. Mertens, H. de Gersem, R.Belmans, K. Hameyer, D. Lahaye, S. Vandewalle,and D. Roose, An Algebraic Multigrid Method for Solving Very Large Electromagnetic Systems, IEEE Trans. Magn. 34, 3327-3330, 1998.
[141] J.D. Moulton, J.E. Dendy, and J.M. Hyman, The Black Box Multigrid Numerical Homogenization Algorithm, J. Comp. Phys. 141, 1-29, 1998
[142] C. Nonino, Using Approximate Inverses in Algebraic Multilevel Methods, Nu-mer. Math. 80, 397-417, 1998.
[143] C. Nonino, Optimal V-Cycle Algebraic Multilevel Preconditioning, Numer. Lin.Alg. Appl. 5, 441-459, 1998.
[144] A.A. Reusken, Multigrid With Matrix-Dependent Transfer Operators for a Singular Perturbation Problem. Computing 50 (3), 199-211, 1993.
[145] J.W. Ruge, Element Interpolation for Algebraic Multigrid (AMG), Presentation at the 4th Copper Mountain Conference on Multigrid Methods, Copper Mountain, CO., 1989.
[146] J.W. Ruge and K. Stuben, Algebraic Multigrid, in Multigrid Methods, S. Mc-Cormick, ed., SIAM, Philadelphia, PA, 1987, pp. 73-130.
[147] K. Stuben, Algebraic Multigrid (AMG): Experiences and comparisons, Appl.Math. Comp. 13, 419-452, 1983.
[148] K. Stuben, A Review of Algebraic Multigrid, GMD Report 69, 1999.
[149] K. Stuben, U. Trottenberg, and K. Witsch, Software Development Based on Multigrid Techniques, in Proc, IFIP-Conference on PDE Software, Modules,Interfaces and Systems, B. Enquist and T. Smedsaas, eds., Soderkoping, Sweden,1983, North-Holland, Amsterdam, 1984, pp. 241-267.
[150] C. Wagner, On the Algebraic Construction of Multilevel Transfer Operators,Computing, 65, 73-95, 2000.
[151] C. Wagner, Introduction to Algebraic Multigrid, Course Notes of an Algebraic Multigrid Course at the University of Heidelberg in the Winter Semester 1998/99.
[152] R. Webster, An Algebraic Multigrid Solve for Navier-Stokes Problems in the Discrete Second Order Approximation, Int. J. Num. Meth. Fluids 22, 1103-1123,1996.
[153] L. Zaslavsky, An Adaptive Algebraic Multigrid for Multigroup Neutron Diffusion Reactor Core Calculations, Appl. Math. Comput. 53, 13-26, 1993.
[154] L. Zaslavsky, An Adaptive Algebraic Multigrid for Reactor Critical Calculations, SIAM J. Sci. Comput. 16, 840-847, 1995.
[155] T.F. Chan and P. Vanek, Multilevel Algebraic Elliptic Solvers, HPCN Europe 1999: 1001-1014.
[156] J. Mandel, M. Brezina, P. Vanek, Energy Optimization of Algebraic Multigrid Bases, Computing 62 (1999), No. 3, 205-228.
[157] P. Vanek, J. Mandel, and M. Brezina, Algebraic Multigrid on Unstructured Meshes, University of Colorado at Denver, UCD/CCM Rep, 34, 1994
[158] W.L. Wan, An Energy-Minimizing Interpolation for Multigrid Methods, Tech.Rep., Department of Mathematics, UCLA, April 1997. UCLA CAM Report 97-18.
[159] W.L. Wan, T.F. Chan, and B. Smith, An Energy-Minimizing Interpolation for Robust Multigrid Methods, SIAM J. Sci. Comput. 21 (2000), No. 4, 1632-1649.
[160] J. Xu and L. Zikatanov, Algebraic multigrid and Numerical Homogenization by Energy-Minimizing Base, to appear.
[161] T. F. Chan, J. Xu and L. Zikatanov, An Agglomeration Multigrid Method for Unstructured Grids, in 10-th International Conference on Domain Decomposition methods, Vol. 218 of Contemporary Mathematics, AMS, 1998, pp. 67-81.
[162] Q. Chang, Y. S. Wong, and Z. Li, New Interpolation Formulas Using Geometric Assumptions in the Algebraic Multigrid Method, Appl. Math. and Comp., 46(1992), pp. 223-254
[163] T. Grauschopf, M. Griebel, H. Regler, Additive Multiscale-Preconditioners Based on Bilinear Interpolation, Matrix Dependent Geometric Coarsening and Algebraic Multigrid Coarsening for Second Order Elliptic PDEs, Appl. Numer. Math. 23, 63-96, 1997.
[164] H. Kim, J. Xu, and L. Zikatanov, A Multigrid Method Based on Graph Matching for Convection Diffusion Equations, Num. Meth. in Lin. Alg., (2002). (to appear).
[165] S. Knapek, Matrix-Dependent Multigrid Homogenization for Diffusion Problems, SIAM J. Sci. Comput. 20 (2), 515-533, 1998.
[166] M. Brezina, A. J. Cleary, R. D. Falgout, V. E. Henson, J. E. Jones, T. A. Manteuffel, S. F. McCormick, and J. W. Ruge. Algebraic Multigrid Based on Element Interpolation (AMGe). SIAM J. Sci. Comput., 22(5): 1570-1592 (electronic), 2000.
[167] V. E. Henson and P. S. Vassilevski, Element-free AMGe: General Algorithms for Computing Interpolation Weights in AMG, SIAM J.Sci Comput., 23 (2001), pp. 629-650 (electronic). Copper Mountain Coference (2000).
[168] P. S. V. J. E. Jones, AMGe Based Element Agglomeration, SIAM J. Sci. Comput., 23 (2001), pp. 109-133 (electronic).
[169] P. S. V. J. E. Jones and P. S. Vassilevski. AMGe Based On Element Agglomeration, SIAM J. Sci. Comp. Vol. 23, No. 1 (1999)
[170] N. A. Pierce, and M. B. Giles, Preconditioned Multigrid Methods for Compressible Flow Calculation on Stretched Meshes, J. Comp. Phys. 136, 425-445, 1997.
[171] Shi shu, Jinchao Xu, Yingxiong Xiao and L.Zikatanov, Algebraic Multigrid Method on Lattice Block Materials, Recent Progress in Computional and Applied PDEs, Edited by Tony.F. Chan, et. al., Boston/London, 289-307, 2002.3, 3-14.
[172] 肖映雄，舒适，张平文，莫则尧，许进超，求解二维三温能量方程的半粗化代数多重网格法，数值计算与计算机应用，No．4(2003)，293—303．
[173] P. Vanek, J. Mandel, and M. Brezina, Algebraic Multigrid by Smoothed Aggregation for Second and Fourth Order Elliptic Problems. Computing, 56 (3): 179-196, 1996.
[174] P. Vanek, M. Brezina, Convergence of Algebraic Multigrid Based on Smoothed Aggregation, Numer. Math., 88 (2001), pp. 559-579.
[175] P. Vanek and J. Krizkova, Two-level Precondition With Small Coarse Grid Appropriate for Unstructured Meshes, Numer. Linear Algebra Appl. 3 (1996), No. 4, 255-274.
[176] J. Xu and L. Zikatanov, The method of Alternating Projections and the Method of Subspace Corrections on Hilbert Space, J. of AMS, 15 (2002), pp. 573-597.
[177] D. N. Arnold and J. Qin, Quadratic velocity/linear pressure stokes element. In G. Richter R. Vichnevetsky D. Knight, editor, Advances in Computer Methods for Partial Differential Equations-Ⅶ. IMACS, 1992.
[178] I. Babuska, Approximation by Hill function, Comment Math. Univ. Carolimae 11(1970), 787-811.
[179] F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Spinger-Verlag, New York-Heidelberg-berlin, 1991.
[180] I. Babuska and T. Strouboulis, FEM-Latest Developments, Open Problems and Perspectives, International Colloquium on Applications of Computer Science and Mathematics in Architecture and Civil Engineering, Weimar, Germany, February 26-March 1, 1997.
[181] I. Babuska and S. A. Sauter, Mathematical description of periodic trusses, in preparation, 1999.
[182] L. J. Gibson and Michael F. Ashby, Cellular Solids Structure and properties, Cambridge University Press, 1997.
[183] 符尚武等，二维三温能量方程的九点差分近似及其迭代解法，计算物理，Vol．15，No．4，1998．
[184] Y. Sadd, Iterative methods for Sparse Linear Systems, PWS Publishing Company, 1995.
[2] P. G. Ciarlet, The Finite Element for Elliptic Problems, North-Holland, Amsterdam, New York, Oxford, 1978.
[3] Chen Chuanmiao, Huang Yunqing, High Accuracy Theory of FEM(in Chinese), Changsha: Hunan Science and Technology Press, 1995.
[4] Huang Yunqing, Shu Shi and Yu Haiyuan, Superconvergence and asymptotic expansions for linear finite element approximations on criss-cross mesh. Science in China Ser. A Mathematics 2004 Vol. 47, Supp. 136-145.
[5] 林群，严宁宁，高效有限元构造与分析，河北大学出版社，1996．
[6] 应隆安，有限元方法讲义，北京大学出版社，1988．
[7] 朱起定，林群，有限元超收敛理论，湖南科学技术出版社，1989．
[8] R. E. Alcouffe, A. Brandt, J.E. Dendy, and J. W. Painter, The Multi-Grid Methods for the Diffusion Equation With Strongly Discontinuous Coefficients, SIAM J. Sci. Stat. Comput. 2 (1981), pp. 430-454.
[9] G. P. Astrakhantsev, An Iterative Method of Solving Elliptic Net Problems, USSR Comp. Math. Math. Phys. 11 (2), 171-182, 1971.
[10] N.S. Bakhvalov, Convergence of a Relaxation Method With Natural Constraints on the Elliptic Operator, USSR, Comp. Math. Math. Phys. 6, 101-135, 1966.
[11] R. Bank, T. Dupont and H. Yserentant, The Hierarchical Basis Multigrid Method. Numer. Math. 52, 427-458, 1988.
[12] R. Bank, and J. Xu, An Algorithm for Coarsing Unstructured Meshes, Numer. Math., 73, No. 1, pp. 1-36, 1996.
[13] F. Bornemann and P. Deuflhard, The Cascadic Multigrid Method for Elliptic Problems, Numer. Math. 75, 135-152, 1996.
[14] D. Braess, The Contraction Number of a Multigrid Method for Solving the Poission Equation. Numer. Math. 37, 387-404, 1981.
[15] D. Braess, W. Hackbusch, A new Convergence Proof for the Multigrid Method Including the V-Cycle. SIAM J. Numer. Anal. 20, 967-975, 1983.
[16] J. H. Bramble, Multigrid Methods, Pitman Research Notes in Mathematical Series 294, Longman, Harlow, 1993.
[17] J. H. Bramble, J. E. Pasciak, J. Wang, and J. Xu, Convergence Estimates for Multigrid Algorithms Without Regularity Assumptions, Math. Comp., 57 (1991), pp. 23-45.
[18] J.H. Bramble, J. E. Pasciak, and J. Xu, Parallel Multilevel Preconditions, Math.Comp. 55 (1990), 1-22.
[19] J.H. Bramble, J. E. Pasciak, and J. Xu, The Analysis of Algorithms With Nonnested Spaces or Noninhereted Quadratic Forms, Math. Comp., 56 (1991),pp. 1-34.
[20] A.Brandt, Multi-level Adaptive Technique (MLAT) for Fast Numerical Solution to Boundary Value Problems, Proceedings of the 3rd International Conference on Numerical Methods in Fluid Mechanics (Paris,1972), Lecture Notes in Physics,18(eds H. Cabannes and R. Temam), Springer-Verlag, Berlin and New York,1973, pp. 82-89.
[21] A.Brandt, Multi-level Adaptive Techniques (MLAT), I. The Multigrid Method,Research Rep. RC 6026, IBM T.J. Watson Research Center, Yorktown Heights,NY, 1976.
[22] A.Brandt, Multi-Level Adaptive Solutions to boundary-Value Problems, Math.Comput. 31, 333-390, 1977.
[23] A.Brandt, Multigrid Techniques: 1984 Guide With Applications to Fluid Dynamics, GMD-Studien Nr. 85, Gesellschaft fur Mathematik und Datenverar-beitung, St.Augustin, 1984.
[24] A. Brandt, Rigorous Quantitative Analysis of Multigrid: I. Constant Coefficients Two Level Cycle With L_2 Norm, SIAM J. Numer. Anal. 31, 1695-1730, 1994.
[25] A.Brandt, B. Diskin, Multigrid Solvers for Non-Aligned Sonic Flows. SIAM J.Sci. Comput. 21, 473-501, 1999.
[26] A.Brandt, and I. Yavneh, On Multigrid Solution of High-Reynolds Incompressible Entering Flows. J. Comp. Phys. 101, 151-164, 1992.
[27] A.Brandt, and I. Yavneh, Accelerated Multigrid Convergence and High Recircu-lating Flows. SIAM J. Sci. Comput. 14, 607-626, 1993.
[28] S. Brenner, Convergence of the Multigrid V-Cycle Algorithm for Second-Order Boundary Value Problems Without Full Elliptice Regularity, Math. Comp. 71(2002), pp. 507-525 (electroic).
[29] W.L. Briggs, A Multigrid Tutorial, Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, 1987 (W.L. Briggs, V.E. Henson, and S.F.McCormick, A Multigrid Tutorial, 2nd Edition, SIAM, Philadelphia, 2000).
[30] C. Buldwin, P.N. Brown, R.D. Falgout, J. Jones, and F. Gramam, Iterative Linear Solvers in a 2D Radiation-Hydrodynamics Code : Methods and Performance,J. Comput. Phys., 154 (1999), pp. 1-40.
[31] R.H. Chan, T.F. Chan, and W.L. Wan, Multigrid for Differential-Convolution Problems Arising From Image Processing, Technical Report CAM 97-20, Department of Mathematics, UCLA, 1997.
[32] T.F. Chan, H.B. Keller, Arc-Length Continuation and Multi-Grid Techniques for Nonlinear Elliptic Eigenvalue Problems, SIAM J. Sci. Comput. 3. 173-194,1982.
[33] T.F. Chan, S. Go, and L. Zikatanov, Multigrid Methods for Elliptic Problems on Unstructured Grids, VKI Lecture Series 1997-02, Van Karman Institute, Rhode St. G. Belgium, 1997.
[34] J.E. Dendy, Jr., M..P. Ida, and J. M. Rutledge. A Semi-Coarsening Multigrid Algorithm for SIMD Machines, SIAM J. Sci. Stat. Comput., 13: 1460-1469,1992.
[35] J.E. Dendy, S.F. McCormick, J.W. Ruge, T.F. Russel, and S. Schaffer, Multi-grid Methods for Three-Dimensional Petroleum Reservoir Simulation, Paper SPE 18409, Tenth Symposium on Reservoir Simulation, Houston, Texas, 1989.
[36] E. Dick, J. Linden, A Multigrid Method for Steady Incompressible Navier-Stokes Equations Based on Flux Difference Splitting, Int. J. Num. Meth. Fluids 14,1311-1323, 1992.
[37] E. Dick, K. Riemslagh, Multi-Staging of Jacobi Relaxation to Improve Smoothing Properties of Multigrid Methods for Steady Euler equations, II. J. Comp. Appl.Math. 59, 339-348, 1995.
[38] E. Dick, and J. Steelant, Coupled Solution of the Steady Compressible Navier-Stokes Equations and the k — e Turbulence Equations With a Multigrid Method,Appl. Numer. 23, 49-61, 1997.
[39] E. Dick, K. Riemslagh, and J. Vierendeels, Multigrid Methods VI, Proceedings of the 6th European Multigrid Conference, Lectured Notes in Computational Science and Engineering 14, Springer, Berlin, 2000.
[40] Y. R. Efendiev, T. Y. Hou, and Z.-H. Wu. Convergence of a Nonconforming Multiscale Finite Element Method. SIAM J. Numer. Anal., 37 (3): 888-910 (electronic), 2000.
[41] B. Engquist and E. Luo, Convergence of a Multigrid Method for Elliptic Equations With Highly Oscillatory Coefficients, SIAM J. Numer. Anal., 34: 2254-2273, 1997.
[42] R.P. Fedorenko, A Relaxation Method for Solving Elliptic Difference Equations,USSR Comp. Math. Math. Phys., 1 (5), 1092-1096, 1962.
[43] R.P.Fedorenko, On the Speed of Convergence of an Iterative Process, USSR Comp. Math. Math. Phys., 4 (3), 227-235, 1964. MR 31#6386.
[44] H. Foerster, K. Stuben, and U. Trottenberg, Non-Standard Multigrid Techniques Using Checkered Relaxation and Intermediate Grids. Elliptic Problem Solvers (ed. M. H. Schultz), 285-300, Academic Press, New York, 1981.
[45] P. O. Frederickson, and O. A. McBryan, Parallel Superconvergent Multigrid, Multigrid Methods: Theory, Applications and Supercomputing (ed. S. Mc-Cormick), 195-210, Marcel Dekker, New York, 1988.
[46] P. O. Frederickson, and O. A. McBryan, Recent Developments for the PSMG Multiscale Method. Multigrid Methods Ⅲ, Proceedings of the 3rd International Conference on Multigrid Methods (eds W. Hackbusch and U. Trottenberg). 21-40, Birkhauser, Basle, 1991.
[47] A. Greenbaum, A Multigrid Method for Multiprocessors, Appl. Math. Comp. 19, 23-45, 1986.
[48] H. Guillard, and P. Vanek, An aggregation Multigrid Solver for Convection-Diffusion Problems on Unstructured Meshes, Rep. 130, Center for Computational Mathematics, University of Denver, 1998.
[49] H. Guillard, and G. W. Zumbusch, Parallel Multigrid in an Adaptive PDE Solver Based on Hashing and Space-Filling Curves, Parallel Comput. 25, 827-843, 1999.
[50] W. Hackbusch, Ein Iteratives Verfahren Zur Schnellen Auflosung Elloptischer Randwertproblem, Rep. 76-12, Institute for Applied Mathematics, University of Cologne, West Germany, 1976.
[51] W. Hackbusch, On the Convergence of a multi-grid iteration applied to finite element equations. Rep. 77-8, Institute for Applied Mathematics, University of Cologne, West Germany, 1977.
[52] W. Hackbusch, On the Multigrid Method Applied to Diffrence Equations, Computing, 20 (1978), pp. 291-306.
[53] W. Hackbusch, Convergence of Multi-Grid Iterations Applied to Difference Equations. Rep. 77-8, Math., Comp. 34. 425-440, 1980.
[54] W. Hackbusch, On the Convergence of a Multi-Grid Iteration. Beitrage Numer. Math. 9, 213-239, 1981
[55] W. Hackbusch, Multigrid Methods and Applications, Vol.4 of Computational Mathematics, Springer-Verlag, Berlin,1985.
[56] W. Hackbusch, The Frequency Decomposition Multigrid Method, part Ⅰ: Appllication to Anisotropic Equations, Numer. Math., 56: 229-245, 1989.
[57] W. Hackbusch and U. Trottenberg (eds), Multigrid Methods, Lecture Notes in Mathematics 960. Springer, Berlin, 1982.
[58] W. Hackbusch and U. Trottenberg (eds), Multigrid Methods II. Lecture Notes in Mathematics 1228. Springer, Berlin, 1986.
[59] W. Hackbusch and U. Trottenberg (eds), Multigrid Methods III. Proceedings of the 3rd International Conference on Multigrid Methods, International Series on Numerical Mathematics 98, Birkhauser, Basle, 1991.
[60] W. Hackbusch and G. Wittum, Multigrid Methods V, Proceedings of the 5th European Multigrid Conference, Lecture Notes in Computational Science and Engineering 3. Springer, Berlin, 1998.
[61] P.W. Hemker, On the Order of Prolongations and Restrictions in Multigrid Procedures, J. Comp. Appl. Math. 32, 423-429, 1990.
[62] P.W. Hemker and P. Wesseling (eds.), Multigrid Methods IV, Proceedings of the 4th European Multigrid Conference. Birkhauser, Basle, 1994.
[63] H. Hoppe, and H. Miihlenbein, Parallel Adaptive Full-Multigrid Methods on Message-Based Multiprocessors, Parallel Comput. 3, 269-287, 1986.
[64] G. Horton, The Time-Parallel Multigrid Method, Comm. Appl. Num. Meth. 8.585-596, 1992.
[65] Yunqing Huang, Zhongci Shi, Tao Tang and Weimin Xue, A multilevel successive iteration method for nonlinear elliptic problems. Math. Comp. 73 (2004), 525-539.
[66] W. Joppich and S. Mijalkovie, Multigrid Methods for Process Simulation.Springer, Vienna, 1993.
[67] R. Kettler, Analysis and Comparison of Relaxation Schemes in Robust Multigrid and Preconditioned Conjugate Gradient Methods, In W. Hackbusch and U. Trottenberg, Editors, Multigrid Methods, pages 502-534, Springer-Verlag, November1981.
[68] H. Kim, J. Xu, and L. Zikatanov, Uniformly Convergent Multigrid Methods for Convection Diffusion Problems Without any Constraint on Coarse Grids, Advanced in Comp. Math., (2002). (to appear).
[69] O. Kolp, and H. Mierendorff, Efficient Multigrid Algorithms for Locally Con-strainted Parallel Systems, Appl. Math. Comp. 19, 169-200, 1986.
[70] B. Koobus, M. H. Lallemand, and A. Dervieux, Unstructured Volume-Agglomeration MG: Solution of the Poisson Equation, International Journal for Numerical Methods in fluids 18 (1994), No.1, 27-42.
[71] F.B. Lin, and F. Sotiropoulos, Strongly-Coupled Multigrid Method for 3D Incompressible Flows Using Near-Wall Turbulence Closures, Trans. ASME J. Fluids Eng. 119, 314-324, 1997.
[72] J. Linden, B. Steckel, K. Stuben, Parallel Multigrid Solution of the Navier-Stokes Equations on General 2D-Domains, Parallel Comput. 7, 461-475, 1988.
[73] J. Linden, G. Lonsdale, H. Ritzdorf, and A. Schuller, Scalability Aspects of Parallel Multigrid. Fut. Generation Comp. Systems 10, 429-439, 1994.
[74] O.A. MaBryan, P.O. frederickson, J. Linden, A. Schuller, K. Solchenbach, K.Stuben, C.A. Thole, U. Trottenberg, Multigrid Methods on Parallel Computers a Survey of Recent Developments, Impact Comput. Sci. Eng. 3, 1-75, 1991.
[75] D.J. Mavripilis, V. Venkatakrishnan, Agglomeration Multigrid for Two-Dimensional Viscous Flows, Comp. Fluids 24, 553-570, 1995.
[76] D.J. Mavripilis, V. Venkatakrishnan, A 3D Agglomeration Multigrid Solver for the Reynolds Averaged Navier-Stokes Equations on Unstructured Meshes, Int.J. Num. Meth. Fluids 23, 527-544, 1996.
[77] S.F. McCormick, Multigrid Methods for variational Problems : Further Results,SIAM J.Numer. Anal. 21 (1984), pp. 255-263.
[78] S.F. McCormick, Multigrid Methods for Variational Problems: General. Theory for the V-Cycle, SIAM J. Numer. Anal. 22(1985), pp. 634-643.
[79] S.F. McCormick, Multigrid Methods. SIAM, Philadelphia, 1988.
[80] S.F. McCormick, and J.W. Ruge, Multigrid Methods for Variational Problems:SIAM J. Numer. Anal. 19 (1982), pp. 924-929.
[81] S.F. McCormick, Multilevel Adaptive Methods for Partial Differential Equations. Frontiers in Applied Mathematics 6. SIAM Philadelphia, 1989.
[82] H. Mierendorff, Parallelization of Multigrid Methods With Local Refinements for a Class of Nonshared Memory Systems, Multigrid Methods: Theory, Applications and Supercomputing (ed. S.F. McCormick), 449-465, Marcel Dekker, New York, 1988.
[83] E. Morano, D.J. Mavripilis and V.C. Venkatakrishnan, Coarsening Strategies for Unstructured Multigrid Techniques With Application to Anisotropic Problems, 7th Copper Mountain Conference on Multigrid Methods, NASA Conference Publication 3339 (eds N.D. Melson, T.A. Manteuffel, S.F. McCormick, and C.C.Douglas), 591-606, NASA, Hampton, 1996.
[84] N.H. Naik, and J. van Rosendale, The Improved Robustness of Multigrid Elliptic Solvers Based on Multiple Semicoarsened Grids, SIAM J. Numer. Anal. 30, 215-229, 1993.
[85] R.A. Nicolaides, On Multiple Grid and Related Techniques for Solving Discrete Elliptic Systems, J. Comp. Phys. 19, 418-431, 1975.
[86] J.E. Pasciak, and J. Xu. Parallel Multilevel Preconditioners, Math. Comp., 55(1990), pp. 1-22.
[87] A.A. Reusken, A Multigrid Method Based on Incomplete Gaussian Elimination,Numer. Linear Algebra Appl. 3 (1996) , No. 5, 369-390
[88] V.V. Shaidurov, Multigrid Methods for Finite Elements, Kluwer, Dordrecht,1995.
[89] Zhong-ci Shi and Xue-jun Xu, Cascadic Multigrid Method for Elliptic Problems,East-West J.Numer.Math., 7 (1999), 199-211.
[90] Zhong-ci Shi and Xue-jun Xu, Cascadic Multigrid for Parabolic Problems, J.Comput. Math. , 18 (2000), 551-560.
[91] Zhong-Ci Shi, Xuejun Xu, A new Cascadic Multigrid, Sciences in China Series 4, Vol. 44, No. 1, pp. 21-30, 2001.
[92] R.A. Smith and A. Weiser, Semicoarsening Multigrid on a Hypercube, SIAM J.Sci. Stat. Comput., 13: 1314-1329, 1992.
[93] S.P. Spekreijse, Multigrid Solution of Second Order Discretizations of Hyperbolic Conservation laws, Math. Comp. 49, 135-155, 1987.
[94] K. Solchenbach, and U. Trottenberg, On the Multigrid Acceleration Approach in Computational Fluid Dynamics, Arbertspapiere der GMD 251, Sankt Augustin,West Germanv, 1987.
[95] C.A. Thole, and U. Trottenberg, A Short Note on Standard Multigrid Algorithms for 3D Problems, Appl. Math. Comp. 27 (2), 101-111, 1988.
[96] U. Trottenberg and A. Schuller, Multigrid, Academic Press Inc., San Diego, CA,2001. With Contributions by A.Brandt,P.Oswald and K.Stuben.
[97] P. Vanek, M. Brezina, and R. Telaur, Two-grid Method for Linear Elasticity on Unstructured Meshes, SIAM J. Sci. Comput., 21 (1999), pp. 900-923.
[98] F. Wang, J. Shen, and J. Xu, A Finite Element Multigrid Preconditioner for Chebyshev-Collocation Methods, Appl. Numer. Math., 33 (2002), pp. 471-477.
[99] T. Washio, C.W. Oosterlee, Krylov Subspace Acceleration for Nonlinear Multi-grid Schemes, Electr. Trans. Num. Anal. 6, 271-290, 1997.
[100] P. Wesseling, An Introduction to Multigrid Methods, John Wiley and Sons,Chichester, 1992.
[101] G. Wittum, Multi-Grid Methods for Stokes and Navier-Stokes Equations With Transforming Smoothers: Algorithms and Numerical results, Numer. Math. 54,543-563, 1989.
[102] J. Wu, H. Ritzdorf, C.W. Oosterlee, B. Steckel, and A. Schuller, Adaptive Parallel Multigrid Solution of 2D Incompressible Navier-Stokes Equations, Int. J. Num. Methods Fluids 24, 875-892, 1997.
[103] J. Xu, Theory of Multilevel Methods, PHD thesis, Cornell University, Ithaca, N. Y., 1989.
[104] J. Xu, Iterative Methods by Space Decomposition and Subspace Correction, SIAM Rev., 34 (1992), pp. 581-613.
[105] J. Xu, A Novel Two-Grid Method for Semilinear Equations, SIAM J. Numer. Anal, 33, (1996), 1759-1777.
[106] J. Xu, The Method of Subspace Corrections, J. Comp. Appl. Math., 128 (2001), pp. 335-362.
[107] J. Xu and A. Zhou, A Two-Grid Discretization Scheme for Eigenvalue Problems, Math. Comp., 70(2001), pp. 17-25.
[108] J. Xu and A. Zhou, Local and Parallel Finite Element Algorithms Based on Two-Grid Discretizations, Math. Comp., 231 (2001), pp. 881-909.
[109] J. Xu and A. Zhou, Locall and Parallel Finite Element Algorithms Based on Two-Grid Discretizations for Nonlinear Problems, Advances in Comp. Math., (to appear)
[110] J. Xu and L. Zikatanov, On Multigrid Methods for Generalized Finite Element Methods, in Proc. Intl. Workshop Meshfree Meth. for PDE, Vol. 26 of Springer Lect. Notes on Comp. Sci. Eng. series, 2002.
[111] I. Yavneh, Multigrid Factors for Red-Black Gauss-Seidel Relaxation Applied to a Class of Elliptic Operators, Numer. Anal. 32, 1126-1138, 1995.
[112] I. Yavneh, On Red-Black SOR Smoothing in Multigrid, SIAM J. Sci. Comput. 17, 180-192, 1996.
[113] I. Yavneh, C.H. Venner, A. Brandt, Fast Multigrid Solution of the Advection Problem With Closed Characteristics, SIAM J. Sci. Comput. 19, 111-125, 1998.
[114] H. Yserentant, On the Multi-Level Splitting of Finite Element Spaces, Numer. Math. 49, 379-412, 1986.
[115] H. Yserentant, Old and new Convergence Proofs for Multigrid Methods. Acta Numerica 2,285-326, 1993.
[116] O. Axelsson, P.S. Vassilevski, Algebraic Multilevel Preconditioning Methods Ⅰ, Num. Math. 56, 157-177, 1989.
[117] D. Braess, Towards Algebraic Multigrid for Ellipitic Problems of Sencond Order, Computing 55 (1995), 379-393
[118] A. Brandt, Algebraic Multigrid Theory: the Symmetric Case, Appl. Math.Comp. , 19, 23-56, 1986
[119] A. Brandt, General Highly Accurate Alegebraic Coarsening Schemes, Proceedings of the 9th Copper Mountain on Multigrid Methods, Copper Mountain, April 1999.
[120] A. Brandt. Multiscale Scientific Computation. Six Year Research Summary.1999.
[121] A. Brandt, Multiscale Scientific Computation: Review 2001, in Multiscale and Multiresolution Methods, Vol. 20 of Lect. Notes Comput. Sci. Eng., Springer,Berlin, 2002, pp. 3-95.
[122] A.Brandt, S.F. McCormick, Algebraic Multigrid (AMG) for Sparse Matrix Equations, in "Sparsity and Its Application", D.J. Evans (ed.), Cambridge University Press, pp. 257-284, Cambridge, 1984.
[123] A. Brandt, S.F. McCormick, and J.W. Ruge, Algebraic multigrid (AMG) for Automatic Multigrid Olution with Application to Geodetic Computations, Institute for Computational Studies, POB 1852, Fort Collins, Colorado, 1982.
[124] A.Brandt, S.F. McCormick, and J.W. Ruge, Multigrid Methods for Differential Eigenproblems, SIAM J. Sci. Stat. Comput., 4 (1983), pp. 244-260.
[125] Qianshun Chang, et al., On the Algebraic Multigrid Method, J. Comput.Physics, 125, 279-292, 1996.
[126] Qianshun Chang, Zhaohui Huang, Efficient Algebraic Multigrid Algorithm and Their Convergence, SIAM J. Sci. Comput. Vol. 24, No. 2, pp. 597-618, 2002.
[127] A.J. Cleary, R.D. Falgout, V.E. Henson, J.E. Jones, Coarse-Grid Selection for Parallel Algebraic Multigrid. Proceedings of the 5th International Symposium on Solving Irregularly Structured Problems in Parallel, Lecture Notes in Computer Science 1457, 104-115, Springer, New York, 1998.
[128] A.J. Cleary, R.D. Falgout, V.E. Henson, J.E. Jones, T.A. Manteuifei, S.F. McCormick, G.N. Miranda, and J.W. Ruge, Robustness and Scalability of Algebraic Multigrid, SIAM J.Sci. Comput. (special Issue on the Fifth Copper Mountain Conference on Iterative Methods, 1998). 21, 1886-1908, 2000.
[129] W. Dahmen, L. Eisner, Algebraic Multigrid Methods and the Schur Complement, Notes on Numerical Fluid Mechanics 23, Vieweg, Braunshweig, 1988.
[130] P.M. de Zeeuw, Matrix-Dependent Prolongations and Restrictions in a Blackbox Multigrid Solver, J. Comput. Appl. Math. 33 (1990), 1-27.
[131] J.E. Dendy, Black Box Multigrid, J. Comput. Phys., 48, 366-386, 1982.
[132] J.E. Dendy, Black Box Multigrid for Nonsymmetric Problems, Appl. Math.Comp. 13, 261-284, 1983.
[133] M. Griebel, T. Neunhoeffer, and H. Regler, Algebraic Multigrid Methods for the Solution of the Navier-Stokes Equations in Complicated Geometries,SFB-Bericht Nr. 342/01/96 A, Institutfur Informatik, Technische Universitat Munchen, Munich, 1996.
[134] R.D. Lonsdale, An Algebraic Multigrid Solver for the Navier-Stokes Equations on Unstructured Meshes, Int. J. Num. Meth. Heat Fluid Flow
[135] W.Z. Huang, Convergence of Algebraic Multigrid Methods for Symmetric Positive Definite Matrices With Weak Diagonal Dominance, Appl. Math, and Comp.,46, pp. 145-164, 1991.
[136] F. Kickinger, Algebraic Multi-Grid for Discrete Elliptic Second Order Problems,Institutsbericht 513, Institut fur Mathematik, Universitat Linz, Austria, 1997.
[137] A. Krechel, K. Stuben, Operator Dependent Interpolation in Algebraic Multi-grid, Multigrid Methods V, Proceedings of the 5th European Multigrid Conference, Lecture Notes in Computational Science and Engineering 3 (eds W.Hackbusch and G. Wittum). 189-211, Springer, Berlin, 1998.
[138] A. Krechel, K. Stuben, Parallel Algebraic Multigrid Based on subdomain Blocking. GMD-Report 71, 1999. Submitted to Parallel Comput.
[139] S.F. McCormick, and J.W. Ruge, Algebraic Multigrid Methods Applied to Problems in Computational Structural Mechanics, State-of-the-Art Surveys on Computational Mechanics, 237-270, ASME, New York, 1989.
[140] R. Mertens, H. de Gersem, R.Belmans, K. Hameyer, D. Lahaye, S. Vandewalle,and D. Roose, An Algebraic Multigrid Method for Solving Very Large Electromagnetic Systems, IEEE Trans. Magn. 34, 3327-3330, 1998.
[141] J.D. Moulton, J.E. Dendy, and J.M. Hyman, The Black Box Multigrid Numerical Homogenization Algorithm, J. Comp. Phys. 141, 1-29, 1998
[142] C. Nonino, Using Approximate Inverses in Algebraic Multilevel Methods, Nu-mer. Math. 80, 397-417, 1998.
[143] C. Nonino, Optimal V-Cycle Algebraic Multilevel Preconditioning, Numer. Lin.Alg. Appl. 5, 441-459, 1998.
[144] A.A. Reusken, Multigrid With Matrix-Dependent Transfer Operators for a Singular Perturbation Problem. Computing 50 (3), 199-211, 1993.
[145] J.W. Ruge, Element Interpolation for Algebraic Multigrid (AMG), Presentation at the 4th Copper Mountain Conference on Multigrid Methods, Copper Mountain, CO., 1989.
[146] J.W. Ruge and K. Stuben, Algebraic Multigrid, in Multigrid Methods, S. Mc-Cormick, ed., SIAM, Philadelphia, PA, 1987, pp. 73-130.
[147] K. Stuben, Algebraic Multigrid (AMG): Experiences and comparisons, Appl.Math. Comp. 13, 419-452, 1983.
[148] K. Stuben, A Review of Algebraic Multigrid, GMD Report 69, 1999.
[149] K. Stuben, U. Trottenberg, and K. Witsch, Software Development Based on Multigrid Techniques, in Proc, IFIP-Conference on PDE Software, Modules,Interfaces and Systems, B. Enquist and T. Smedsaas, eds., Soderkoping, Sweden,1983, North-Holland, Amsterdam, 1984, pp. 241-267.
[150] C. Wagner, On the Algebraic Construction of Multilevel Transfer Operators,Computing, 65, 73-95, 2000.
[151] C. Wagner, Introduction to Algebraic Multigrid, Course Notes of an Algebraic Multigrid Course at the University of Heidelberg in the Winter Semester 1998/99.
[152] R. Webster, An Algebraic Multigrid Solve for Navier-Stokes Problems in the Discrete Second Order Approximation, Int. J. Num. Meth. Fluids 22, 1103-1123,1996.
[153] L. Zaslavsky, An Adaptive Algebraic Multigrid for Multigroup Neutron Diffusion Reactor Core Calculations, Appl. Math. Comput. 53, 13-26, 1993.
[154] L. Zaslavsky, An Adaptive Algebraic Multigrid for Reactor Critical Calculations, SIAM J. Sci. Comput. 16, 840-847, 1995.
[155] T.F. Chan and P. Vanek, Multilevel Algebraic Elliptic Solvers, HPCN Europe 1999: 1001-1014.
[156] J. Mandel, M. Brezina, P. Vanek, Energy Optimization of Algebraic Multigrid Bases, Computing 62 (1999), No. 3, 205-228.
[157] P. Vanek, J. Mandel, and M. Brezina, Algebraic Multigrid on Unstructured Meshes, University of Colorado at Denver, UCD/CCM Rep, 34, 1994
[158] W.L. Wan, An Energy-Minimizing Interpolation for Multigrid Methods, Tech.Rep., Department of Mathematics, UCLA, April 1997. UCLA CAM Report 97-18.
[159] W.L. Wan, T.F. Chan, and B. Smith, An Energy-Minimizing Interpolation for Robust Multigrid Methods, SIAM J. Sci. Comput. 21 (2000), No. 4, 1632-1649.
[160] J. Xu and L. Zikatanov, Algebraic multigrid and Numerical Homogenization by Energy-Minimizing Base, to appear.
[161] T. F. Chan, J. Xu and L. Zikatanov, An Agglomeration Multigrid Method for Unstructured Grids, in 10-th International Conference on Domain Decomposition methods, Vol. 218 of Contemporary Mathematics, AMS, 1998, pp. 67-81.
[162] Q. Chang, Y. S. Wong, and Z. Li, New Interpolation Formulas Using Geometric Assumptions in the Algebraic Multigrid Method, Appl. Math. and Comp., 46(1992), pp. 223-254
[163] T. Grauschopf, M. Griebel, H. Regler, Additive Multiscale-Preconditioners Based on Bilinear Interpolation, Matrix Dependent Geometric Coarsening and Algebraic Multigrid Coarsening for Second Order Elliptic PDEs, Appl. Numer. Math. 23, 63-96, 1997.
[164] H. Kim, J. Xu, and L. Zikatanov, A Multigrid Method Based on Graph Matching for Convection Diffusion Equations, Num. Meth. in Lin. Alg., (2002). (to appear).
[165] S. Knapek, Matrix-Dependent Multigrid Homogenization for Diffusion Problems, SIAM J. Sci. Comput. 20 (2), 515-533, 1998.
[166] M. Brezina, A. J. Cleary, R. D. Falgout, V. E. Henson, J. E. Jones, T. A. Manteuffel, S. F. McCormick, and J. W. Ruge. Algebraic Multigrid Based on Element Interpolation (AMGe). SIAM J. Sci. Comput., 22(5): 1570-1592 (electronic), 2000.
[167] V. E. Henson and P. S. Vassilevski, Element-free AMGe: General Algorithms for Computing Interpolation Weights in AMG, SIAM J.Sci Comput., 23 (2001), pp. 629-650 (electronic). Copper Mountain Coference (2000).
[168] P. S. V. J. E. Jones, AMGe Based Element Agglomeration, SIAM J. Sci. Comput., 23 (2001), pp. 109-133 (electronic).
[169] P. S. V. J. E. Jones and P. S. Vassilevski. AMGe Based On Element Agglomeration, SIAM J. Sci. Comp. Vol. 23, No. 1 (1999)
[170] N. A. Pierce, and M. B. Giles, Preconditioned Multigrid Methods for Compressible Flow Calculation on Stretched Meshes, J. Comp. Phys. 136, 425-445, 1997.
[171] Shi shu, Jinchao Xu, Yingxiong Xiao and L.Zikatanov, Algebraic Multigrid Method on Lattice Block Materials, Recent Progress in Computional and Applied PDEs, Edited by Tony.F. Chan, et. al., Boston/London, 289-307, 2002.3, 3-14.
[172] 肖映雄，舒适，张平文，莫则尧，许进超，求解二维三温能量方程的半粗化代数多重网格法，数值计算与计算机应用，No．4(2003)，293—303．
[173] P. Vanek, J. Mandel, and M. Brezina, Algebraic Multigrid by Smoothed Aggregation for Second and Fourth Order Elliptic Problems. Computing, 56 (3): 179-196, 1996.
[174] P. Vanek, M. Brezina, Convergence of Algebraic Multigrid Based on Smoothed Aggregation, Numer. Math., 88 (2001), pp. 559-579.
[175] P. Vanek and J. Krizkova, Two-level Precondition With Small Coarse Grid Appropriate for Unstructured Meshes, Numer. Linear Algebra Appl. 3 (1996), No. 4, 255-274.
[176] J. Xu and L. Zikatanov, The method of Alternating Projections and the Method of Subspace Corrections on Hilbert Space, J. of AMS, 15 (2002), pp. 573-597.
[177] D. N. Arnold and J. Qin, Quadratic velocity/linear pressure stokes element. In G. Richter R. Vichnevetsky D. Knight, editor, Advances in Computer Methods for Partial Differential Equations-Ⅶ. IMACS, 1992.
[178] I. Babuska, Approximation by Hill function, Comment Math. Univ. Carolimae 11(1970), 787-811.
[179] F. Brezzi and M. Fortin, Mixed and hybrid finite element methods, Spinger-Verlag, New York-Heidelberg-berlin, 1991.
[180] I. Babuska and T. Strouboulis, FEM-Latest Developments, Open Problems and Perspectives, International Colloquium on Applications of Computer Science and Mathematics in Architecture and Civil Engineering, Weimar, Germany, February 26-March 1, 1997.
[181] I. Babuska and S. A. Sauter, Mathematical description of periodic trusses, in preparation, 1999.
[182] L. J. Gibson and Michael F. Ashby, Cellular Solids Structure and properties, Cambridge University Press, 1997.
[183] 符尚武等，二维三温能量方程的九点差分近似及其迭代解法，计算物理，Vol．15，No．4，1998．
[184] Y. Sadd, Iterative methods for Sparse Linear Systems, PWS Publishing Company, 1995.