求解三维弹性力学问题高次有限元方程的代数多层网格法
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摘要
本文主要研究求解三维弹性力学问题的高次有限元方程的代数多层网格(AMG)法。首先，给出了一种常用AMG法，并将其应用于三维线弹性问题高次有限元方程的求解，数值结果表明该方法对高次元的应用效果要差于线性元情形。然后，通过分析线性有限元空间和高次有限元空间基函数之间的内在关系，给出了三维二次和三次有限元空间的网格粗化算法和限制算子的构造方法，进而得到求解线弹性问题高次有限元方程的AMG法，同时对其收敛性给出了严格的理论证明。新的AMG法从本质上克服了粗网格层自由度难以控制等通常AMG法的缺陷。另外，我们还针对线弹性问题给出了基于块对角逆的预条件子的PCG方法。数值实验结果表明，本文所建立的AMG法和PCG方法对求解三维线弹性问题高次有限元方程具有高效性和鲁棒性(robustness)。
In this paper, we discuss algebraic multigrid (AMG) methods for higher-order finite element equations in three dimensional linear elasticity. First, we give a commonly used AMG method, and apply it to higher-order finite element equations in three dimensional linear elasticity. Numerical results show that this AMG method is efficient for the linear element, but inefficient for the higher-order elements. By analyzing the relationship between the linear finite element space and the higher-order finite element space, we then obtain a new coarsening algorithm and the corresponding interpolation operator, and apply the AMG method to solving quadratic and cubic Lagrangian finite element discretization systems arising from three dimensional linear elasticity. At the same time we present the rigorous theoretical analysis of convergence for the resulting AMG methods. The new methods overcome one common difficulty in the commonly used AMG method, i.e., the number of coarse grid degrees of freedom is not easy to control. In addition, we present the PCG method for higher-order finite element equations in three dimensional linear elasticity by choosing the inverses of block diagonal matrixes as a preconditioner. Numerical results show that the constructed AMG methods and PCG method are robust and efficient for solving the higher-order finite element equations in three dimensional linear elasticity.
In this paper, we discuss algebraic multigrid (AMG) methods for higher-order finite element equations in three dimensional linear elasticity. First, we give a commonly used AMG method, and apply it to higher-order finite element equations in three dimensional linear elasticity. Numerical results show that this AMG method is efficient for the linear element, but inefficient for the higher-order elements. By analyzing the relationship between the linear finite element space and the higher-order finite element space, we then obtain a new coarsening algorithm and the corresponding interpolation operator, and apply the AMG method to solving quadratic and cubic Lagrangian finite element discretization systems arising from three dimensional linear elasticity. At the same time we present the rigorous theoretical analysis of convergence for the resulting AMG methods. The new methods overcome one common difficulty in the commonly used AMG method, i.e., the number of coarse grid degrees of freedom is not easy to control. In addition, we present the PCG method for higher-order finite element equations in three dimensional linear elasticity by choosing the inverses of block diagonal matrixes as a preconditioner. Numerical results show that the constructed AMG methods and PCG method are robust and efficient for solving the higher-order finite element equations in three dimensional linear elasticity.
引文
[1] A.Brandt. Multi-level adaptive solutions to boundary-value problems[J]. Math. Comp., 31(1977): 333-390.
[2] W.Hackbusch. Multigrid Methods and Applications[J]. Vol.4 of Computational Mathematics, Springer-Verlag, Berlin, 1985.
[3] I. D. Parsons and J, F. Hall. The Multigrid Method in Solid Mechanics[J]. Intl. J. for Numer. Meth. in Eng., 1990, 29: 719-754.
[4] E. Dick, J. Linden. A Multigrid Method for Steady Incompressible Navier-Stokes Equations Based on Flux Difference Splitting[J]. Int. J. Num. Meth. Fluids 14,1311-1323, 1992.
[5] R. H. Chart, T. F. Chart, and W.L. Wan. Multigrid for Differential-Convolution Problems Arising From image Processing[J]. Technical Report CAM 97-20, Department of Mathematics, UCLA, 1997.
[6] A. Brandt, and I. Yavneh. On Multigrid Solution of High-Reynolds Incompressible Entering Flows[J]. J. Comp. Phys. 101, 151-164, 1992.
[7] J. Ruge. AMG for higher-order discretizations of second-order elliptic problems[EB], in: Abstracts of the Eleventh Copper Mountain Conference on Multigrid Methods (http://www.mgnet.org/mgnet /Conferences/Copper Mtn03/Talks/ruge.pdf), 2003.
[8] J. J. Heys, T.A.Manteuffel, S.F.Mcormick and L.N.Olson. Algebraic Multigrid (AMG) for Higher-Order Finite Elements[J].Journal of Computational Physics, 204(2005): 520-532.
[9] 孙杜杜，舒适．求解三维高次拉格朗日有限元方程的代数多重网格法[J]．计算数学，27(2005)：101-112．
[10] Shi Shu, Sun Dudu and Jinchao Xu. An Algebraic Multigrids for Higher Order Finite Element discretiztions[J]. Computing 77, 347-377(2006).
[11] Liu Xuan,Dudu Sun,Shi Shu．求解非结构四边形二次拉格朗日有限元方程的代数多重网格法[J]．湘潭大学自然科学学报，2005，27(3)：8-15．
[12] Yunqing HUANG, Shi SHU,Xijun YU. Preconditioning Higher Order Finite Element Systems by Algebraic Multigrid Method of Linear Elements[J]. J.Comp.Math.,24(2006)5: 657-664.
[13] A. Brandt. Algebraic Multigrid Theory: the Symmetric Case[J]. Appl. Math: Comp., 19, 23-56, 1986.
[14] J.Ruge and K. Stuben. Algebraic multigrid in Multigrid Methods[M]. S. McCormick, ed., SIAM, Philadelphia, PA, 1987.
[15] Q.S.Chang,Y.S.wong and H.Fu. On the Algebraic Multigrid Methods[J]. J.Comput. Phys.,125: 279-292,1996.
[16] K. Stuben. Algebraic Multigrid (AMG): Experiences and comparisons[J]. Appl. Math. Comp. 13, 419-452, 1983.
[17] J. Mandel, M.Brezina, P.Vanek. Energy optimization of algebraic multigrid bases[J]. Comput., 1999, 62(3): 205-228.
[18] W. L. Wan, T. F. Chan, and B. Smith. An energy-minimizing interpolation for robust multigrid methods[J]. SIAM J. Sci. Comput. 21(2000), No. 3, 559-579.
[19] J. Xu and L.Zikatanov. On the energy minimizing base for algebraic multigrid methods[J]. Comput. Visual Sci., 2004.
[20] P. Vanek, J.Mandel, and M.Brezina. Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems[J]. Computing, 56(3): 179-196, 1996.
[21] P. Vanek, M. Brezina, and J. Mandel. Convergence of algebraic multigrid based on smoothed aggregation[J]. Numer. Math., 88(3): 559-579, 2001.
[22] M. Griebel, D. Oeltz, and M. A. Schweitzer. An algebraic multigrid method for linear elasticity[J]. SIAM J. Sci. Comput., 25(2): 385-407, 2003.
[23] M. Brezina, A. J. Cleary, R. D. Faigout, V. E. Henson, J. E. Jones, T. A. Manteuffel, S. F. McCormick, and J. W. Ruge. Algebraic multigrid based on element interpolation (AMGe)[J]. SIAM J. Sci. Comput., 22(5): 1570-1592, 2000.
[24] Qiang Du, Zhaohui Huang, and Desheng Wang. Mesh and Solver Coadaptation in Finite Element Methods for Anisotropic Problems, Numerical Methods for Differential Equations[J]. 2005,21: 859-874.
[25] P.Vanek, J.Mandel,and M.Brezina. Algebraic multigrid on unstructured meshes[R]. UCD/CCM Report 34,Center for Computational Mathematics ,University of Colorado at Denver,December 1994.
[26] T. F. Chart, J. Xu and L. Zikatanov. An Agglomeration Multigrid Method for Unstructured Grids[J]. in 10-th International Conference on Domain Decomposition methods, Vol. 218 of Contemporary Mathematics, AMS, 1998, pp. 67-81.
[27] J. E. Jones and P.S.Vassilevski. AMGe based on agglomeration[J]. SIAM J.Sci.Comput. 23: 109-133, 2001.
[28] V. E. Henson and P.S.Vassilevski. Element-free AMGe: General algorithms for computing interpolation weights in AMG[J]. SIAM J.Sci.Comput.23: 629-659, 2001.
[29] T. Chartier, R.D.Falgout, V.E.Henson, J.E. Jones, T.Manteuffel, S.F Mccormick, J. W Ruge and P.S.Vassilevski. Spectral AMGe( AMGe)[J]. SIAM J.Sci.Comp. 20(1): 1-20, 2003.
[30] J. E.Dendy Jr., M.P.Ida, J.M.Rutledge. A semi-coarsening multigrid algorithm for SIMD machines[J]. SIAM J.Sci.Statist.Comput.13(1992)1460-1469.
[31] J. Xu and L. Zikatanov. The method of Alternating Projections and the Method of Subspace Corrections on Hilbert Space[J]. Journal of the American Mathematics Society, 15 (2002): 573-597.
[32] Alan A. Powell, F. H. G. Gruen. The Constant Elasticity of Transformation Production Frontier and Linear Supply System [J]. International Economic Review, Vol. 9, No. 3 (Oct., 1968), pp. 315-328.
[33] Timoshenko, Stephen P, Goodier, J. N. Theory of elasticity[M] Engineering Societies Monographs, New York: McGraw-Hill, 1970, 3rd ed.
[34] 徐芝纶，弹性力学[M]．1990．北京：高等教育出版社．
[35] Ting, T. C. T. Anisotropic Elasticity: Theory And Applications [M]. Oxford Engineering Science Series,580 pages Published: April 1996.
[36] A. H. Krall and D. A. Weitz. Internal Dynamics and Elasticity of Fractal Colloidal Gels, Phys[J]. Rev. Lett. 80, 778-781 (1998) [Issue 4-26 January 1998 ].
[37] B. Smith. Domain Decomposition Algorithms for the Partial Differential Equations of Linear Elasticity[R]. Technical Report: TR1990-517.
[38] Richard S. Falk . Nonconforming Finite Element Methods for the Equations of Linear Elasticity [J]. Mathematics of Computation, Vol. 57, No. 196 (Oct., 1991), pp. 529-550
[39] Barry F. Smith . An optimal domain decomposition preconditioner for the finite element solution of linear elasticity problems,SIAM Journal on Scientific and Statistical Computing [J]. Volume 13, Issue 1 (January 1992).
[40] Susanne C. Brenner, Li-Yeng Sung. Linear Finite Element Methods for Planar Linear Elasticity [J]. Mathematics of Computation, Vol. 59, No. 200 (Oct., 1992), pp. 321-338.
[41] Susanne C. Brenner. A Nonconforming Mixed Multigrid Method for the Pure Displacement Problem in Planar Linear Elasticity [J]. SIAM Journal on Numerical Analysis, Vol. 30, No. 1 (Feb., 1993), pp. 116-135.
[42] R. Rannacher and F.-T. Suttmeier . A feed-back approach to error control in finite element methods: application to linear elasticity [J]. Computational Mechanics, Issue: Volume 19, Number 5, Date: April 1997 Pages: 434-446.
[43] Zhiqiang Cai, Thomas A. Manteuffel, Stephen F. McCormick, Seymour V. Parter. FirstOrder System Least Squares (FOSLS) for Planar Linear Elasticity: Pure Traction Problem [J]. SINUM, Volume 35 Issue 1 , pages 320-335.
[44] P. Vanek, M. Brezina and R. Tezaur. Two-grid Method for Linear Elasticity on Unstructured Meshes[J]. SIAM J. Sci. Comput., 21, 3(1999), 900-923.
[45] O. Axelsson and A. Padiy. On a two level Newton type procedure applied for solving nonlinear elasticity problems [J]. Int. J. Numer. Meth. Engng. 49 (2000), 1479-1493.
[46] Isaac Fried. A gradient computational procedure for the solution of large problems arising from the finite element discretization method [J]. Journal of the American Mathematics Society, International Journal for Numerical Methods in Engineering, Volume 2, Issue 4, Pages 477-494.
[47] Axel Klawonn, Olof B. Widlund. A Domain Decomposition Method with Lagrange Multipliers and Inexact Solvers for Linear Elasticity [J]. SISC, Volume 22 Issue 4, pages 1199-12191 2000 Society for Industrial and Applied Mathematics.
[48] Ralf Kornhuber and Rolf Krause. Adaptive multigrid methods for Signorini's problem in linear elasticity [J]. Computing and Visualization in Science, ISSN: 1432-9360 (Paper) 1433-0369 (Online).
[49] 罗振东．混合有限元法基本理论及应用[M]．山东省教育出版社．1996．
[50] 赵钍焱．求解二维弹性力学问题高次有限元方程的代数多重网格方法[D]．湘潭大学图书馆，2006．
[51] 肖映雄，张平，舒适，阳莺．等代数结构面网格剖分下三维弹性问题的代数多重网格法[J]．工程力学，2005，22(6)：76-81．
[52] Shi Shu, Jinchao Xu, Ying Yang, and Haiyuan Yu. An algebraic multigrid method for finite element systems on criss-cross grids [J]. Advances in Computational Mathematics, 25(2006): 287-304.
[53] 舒适，黄云清，阳莺，蔚喜军．一类三维等代数结构面网格剖分下的代数多重网格法[J]．计算物理，22：6(2005)，480-488．
[54] 谭敏，肖映雄，舒适．一种各向异性四边形网格下的代数多重网格法[J]．湘潭大学自然科学学报．2005，27(1)：78-84．
[55] R.E.Caflisch, Y. J. Lee, S. Shu, Y. X. Xiao, J. Xu. An application of multigrid methods for a discrete elastic model for epitaxial systems [J]. J. Comp. Phys., 219(2006): 697-714.
[56] Yingxiong Xiao, Ping Zhang and Shi Shu(通讯作者). An algebraic multigrid method with interpolation reproducing rigid body modes for semi-definite problems in two dimensional linear elasticity [J]. J. Comp. Appl. Math., 2007, 200(2): 637-652.
[57] Ying-Xiong Xiao, Ping Zhang and Shi Shu. Algebraic multigrid algorithms and numerical simulation for elastic structures with highly discontinuous coefficients [EB]. Math. Comput. Simul (online, www. Science direct.com/science/journal/0378-4754.)
[58] Scott MacLachlan. Adaptive multigrid methods for heterogeneous problems [EB]. http://www-users.cs.umn.edu/maclach/research/SMdelft2006.pdf, edu June 21, 2006.
[2] W.Hackbusch. Multigrid Methods and Applications[J]. Vol.4 of Computational Mathematics, Springer-Verlag, Berlin, 1985.
[3] I. D. Parsons and J, F. Hall. The Multigrid Method in Solid Mechanics[J]. Intl. J. for Numer. Meth. in Eng., 1990, 29: 719-754.
[4] E. Dick, J. Linden. A Multigrid Method for Steady Incompressible Navier-Stokes Equations Based on Flux Difference Splitting[J]. Int. J. Num. Meth. Fluids 14,1311-1323, 1992.
[5] R. H. Chart, T. F. Chart, and W.L. Wan. Multigrid for Differential-Convolution Problems Arising From image Processing[J]. Technical Report CAM 97-20, Department of Mathematics, UCLA, 1997.
[6] A. Brandt, and I. Yavneh. On Multigrid Solution of High-Reynolds Incompressible Entering Flows[J]. J. Comp. Phys. 101, 151-164, 1992.
[7] J. Ruge. AMG for higher-order discretizations of second-order elliptic problems[EB], in: Abstracts of the Eleventh Copper Mountain Conference on Multigrid Methods (http://www.mgnet.org/mgnet /Conferences/Copper Mtn03/Talks/ruge.pdf), 2003.
[8] J. J. Heys, T.A.Manteuffel, S.F.Mcormick and L.N.Olson. Algebraic Multigrid (AMG) for Higher-Order Finite Elements[J].Journal of Computational Physics, 204(2005): 520-532.
[9] 孙杜杜，舒适．求解三维高次拉格朗日有限元方程的代数多重网格法[J]．计算数学，27(2005)：101-112．
[10] Shi Shu, Sun Dudu and Jinchao Xu. An Algebraic Multigrids for Higher Order Finite Element discretiztions[J]. Computing 77, 347-377(2006).
[11] Liu Xuan,Dudu Sun,Shi Shu．求解非结构四边形二次拉格朗日有限元方程的代数多重网格法[J]．湘潭大学自然科学学报，2005，27(3)：8-15．
[12] Yunqing HUANG, Shi SHU,Xijun YU. Preconditioning Higher Order Finite Element Systems by Algebraic Multigrid Method of Linear Elements[J]. J.Comp.Math.,24(2006)5: 657-664.
[13] A. Brandt. Algebraic Multigrid Theory: the Symmetric Case[J]. Appl. Math: Comp., 19, 23-56, 1986.
[14] J.Ruge and K. Stuben. Algebraic multigrid in Multigrid Methods[M]. S. McCormick, ed., SIAM, Philadelphia, PA, 1987.
[15] Q.S.Chang,Y.S.wong and H.Fu. On the Algebraic Multigrid Methods[J]. J.Comput. Phys.,125: 279-292,1996.
[16] K. Stuben. Algebraic Multigrid (AMG): Experiences and comparisons[J]. Appl. Math. Comp. 13, 419-452, 1983.
[17] J. Mandel, M.Brezina, P.Vanek. Energy optimization of algebraic multigrid bases[J]. Comput., 1999, 62(3): 205-228.
[18] W. L. Wan, T. F. Chan, and B. Smith. An energy-minimizing interpolation for robust multigrid methods[J]. SIAM J. Sci. Comput. 21(2000), No. 3, 559-579.
[19] J. Xu and L.Zikatanov. On the energy minimizing base for algebraic multigrid methods[J]. Comput. Visual Sci., 2004.
[20] P. Vanek, J.Mandel, and M.Brezina. Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems[J]. Computing, 56(3): 179-196, 1996.
[21] P. Vanek, M. Brezina, and J. Mandel. Convergence of algebraic multigrid based on smoothed aggregation[J]. Numer. Math., 88(3): 559-579, 2001.
[22] M. Griebel, D. Oeltz, and M. A. Schweitzer. An algebraic multigrid method for linear elasticity[J]. SIAM J. Sci. Comput., 25(2): 385-407, 2003.
[23] M. Brezina, A. J. Cleary, R. D. Faigout, V. E. Henson, J. E. Jones, T. A. Manteuffel, S. F. McCormick, and J. W. Ruge. Algebraic multigrid based on element interpolation (AMGe)[J]. SIAM J. Sci. Comput., 22(5): 1570-1592, 2000.
[24] Qiang Du, Zhaohui Huang, and Desheng Wang. Mesh and Solver Coadaptation in Finite Element Methods for Anisotropic Problems, Numerical Methods for Differential Equations[J]. 2005,21: 859-874.
[25] P.Vanek, J.Mandel,and M.Brezina. Algebraic multigrid on unstructured meshes[R]. UCD/CCM Report 34,Center for Computational Mathematics ,University of Colorado at Denver,December 1994.
[26] T. F. Chart, J. Xu and L. Zikatanov. An Agglomeration Multigrid Method for Unstructured Grids[J]. in 10-th International Conference on Domain Decomposition methods, Vol. 218 of Contemporary Mathematics, AMS, 1998, pp. 67-81.
[27] J. E. Jones and P.S.Vassilevski. AMGe based on agglomeration[J]. SIAM J.Sci.Comput. 23: 109-133, 2001.
[28] V. E. Henson and P.S.Vassilevski. Element-free AMGe: General algorithms for computing interpolation weights in AMG[J]. SIAM J.Sci.Comput.23: 629-659, 2001.
[29] T. Chartier, R.D.Falgout, V.E.Henson, J.E. Jones, T.Manteuffel, S.F Mccormick, J. W Ruge and P.S.Vassilevski. Spectral AMGe( AMGe)[J]. SIAM J.Sci.Comp. 20(1): 1-20, 2003.
[30] J. E.Dendy Jr., M.P.Ida, J.M.Rutledge. A semi-coarsening multigrid algorithm for SIMD machines[J]. SIAM J.Sci.Statist.Comput.13(1992)1460-1469.
[31] J. Xu and L. Zikatanov. The method of Alternating Projections and the Method of Subspace Corrections on Hilbert Space[J]. Journal of the American Mathematics Society, 15 (2002): 573-597.
[32] Alan A. Powell, F. H. G. Gruen. The Constant Elasticity of Transformation Production Frontier and Linear Supply System [J]. International Economic Review, Vol. 9, No. 3 (Oct., 1968), pp. 315-328.
[33] Timoshenko, Stephen P, Goodier, J. N. Theory of elasticity[M] Engineering Societies Monographs, New York: McGraw-Hill, 1970, 3rd ed.
[34] 徐芝纶，弹性力学[M]．1990．北京：高等教育出版社．
[35] Ting, T. C. T. Anisotropic Elasticity: Theory And Applications [M]. Oxford Engineering Science Series,580 pages Published: April 1996.
[36] A. H. Krall and D. A. Weitz. Internal Dynamics and Elasticity of Fractal Colloidal Gels, Phys[J]. Rev. Lett. 80, 778-781 (1998) [Issue 4-26 January 1998 ].
[37] B. Smith. Domain Decomposition Algorithms for the Partial Differential Equations of Linear Elasticity[R]. Technical Report: TR1990-517.
[38] Richard S. Falk . Nonconforming Finite Element Methods for the Equations of Linear Elasticity [J]. Mathematics of Computation, Vol. 57, No. 196 (Oct., 1991), pp. 529-550
[39] Barry F. Smith . An optimal domain decomposition preconditioner for the finite element solution of linear elasticity problems,SIAM Journal on Scientific and Statistical Computing [J]. Volume 13, Issue 1 (January 1992).
[40] Susanne C. Brenner, Li-Yeng Sung. Linear Finite Element Methods for Planar Linear Elasticity [J]. Mathematics of Computation, Vol. 59, No. 200 (Oct., 1992), pp. 321-338.
[41] Susanne C. Brenner. A Nonconforming Mixed Multigrid Method for the Pure Displacement Problem in Planar Linear Elasticity [J]. SIAM Journal on Numerical Analysis, Vol. 30, No. 1 (Feb., 1993), pp. 116-135.
[42] R. Rannacher and F.-T. Suttmeier . A feed-back approach to error control in finite element methods: application to linear elasticity [J]. Computational Mechanics, Issue: Volume 19, Number 5, Date: April 1997 Pages: 434-446.
[43] Zhiqiang Cai, Thomas A. Manteuffel, Stephen F. McCormick, Seymour V. Parter. FirstOrder System Least Squares (FOSLS) for Planar Linear Elasticity: Pure Traction Problem [J]. SINUM, Volume 35 Issue 1 , pages 320-335.
[44] P. Vanek, M. Brezina and R. Tezaur. Two-grid Method for Linear Elasticity on Unstructured Meshes[J]. SIAM J. Sci. Comput., 21, 3(1999), 900-923.
[45] O. Axelsson and A. Padiy. On a two level Newton type procedure applied for solving nonlinear elasticity problems [J]. Int. J. Numer. Meth. Engng. 49 (2000), 1479-1493.
[46] Isaac Fried. A gradient computational procedure for the solution of large problems arising from the finite element discretization method [J]. Journal of the American Mathematics Society, International Journal for Numerical Methods in Engineering, Volume 2, Issue 4, Pages 477-494.
[47] Axel Klawonn, Olof B. Widlund. A Domain Decomposition Method with Lagrange Multipliers and Inexact Solvers for Linear Elasticity [J]. SISC, Volume 22 Issue 4, pages 1199-12191 2000 Society for Industrial and Applied Mathematics.
[48] Ralf Kornhuber and Rolf Krause. Adaptive multigrid methods for Signorini's problem in linear elasticity [J]. Computing and Visualization in Science, ISSN: 1432-9360 (Paper) 1433-0369 (Online).
[49] 罗振东．混合有限元法基本理论及应用[M]．山东省教育出版社．1996．
[50] 赵钍焱．求解二维弹性力学问题高次有限元方程的代数多重网格方法[D]．湘潭大学图书馆，2006．
[51] 肖映雄，张平，舒适，阳莺．等代数结构面网格剖分下三维弹性问题的代数多重网格法[J]．工程力学，2005，22(6)：76-81．
[52] Shi Shu, Jinchao Xu, Ying Yang, and Haiyuan Yu. An algebraic multigrid method for finite element systems on criss-cross grids [J]. Advances in Computational Mathematics, 25(2006): 287-304.
[53] 舒适，黄云清，阳莺，蔚喜军．一类三维等代数结构面网格剖分下的代数多重网格法[J]．计算物理，22：6(2005)，480-488．
[54] 谭敏，肖映雄，舒适．一种各向异性四边形网格下的代数多重网格法[J]．湘潭大学自然科学学报．2005，27(1)：78-84．
[55] R.E.Caflisch, Y. J. Lee, S. Shu, Y. X. Xiao, J. Xu. An application of multigrid methods for a discrete elastic model for epitaxial systems [J]. J. Comp. Phys., 219(2006): 697-714.
[56] Yingxiong Xiao, Ping Zhang and Shi Shu(通讯作者). An algebraic multigrid method with interpolation reproducing rigid body modes for semi-definite problems in two dimensional linear elasticity [J]. J. Comp. Appl. Math., 2007, 200(2): 637-652.
[57] Ying-Xiong Xiao, Ping Zhang and Shi Shu. Algebraic multigrid algorithms and numerical simulation for elastic structures with highly discontinuous coefficients [EB]. Math. Comput. Simul (online, www. Science direct.com/science/journal/0378-4754.)
[58] Scott MacLachlan. Adaptive multigrid methods for heterogeneous problems [EB]. http://www-users.cs.umn.edu/maclach/research/SMdelft2006.pdf, edu June 21, 2006.