A geometric-based algebraic multigrid method for higher-order finite element equations in two-dimensional linear elasticity
In this paper, we will discuss the geometric‐based algebraic multigrid (AMG) method for two‐dimensional linear elasticity problems discretized using quadratic and cubic elements. First, a two‐level method is proposed by analyzing the relationship between the linear finite element space and higher‐or...
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| Published in | Numerical linear algebra with applications Vol. 16; no. 7; pp. 535 - 559 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Chichester, UK
John Wiley & Sons, Ltd
01.07.2009
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| Subjects | |
| Online Access | Get full text |
| ISSN | 1070-5325 1099-1506 |
| DOI | 10.1002/nla.629 |
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| Abstract | In this paper, we will discuss the geometric‐based algebraic multigrid (AMG) method for two‐dimensional linear elasticity problems discretized using quadratic and cubic elements. First, a two‐level method is proposed by analyzing the relationship between the linear finite element space and higher‐order finite element space. And then a geometric‐based AMG method is obtained with the existing solver used as a solver on the first coarse level. The resulting AMG method is applied to some typical elasticity problems including the plane strain problem with jumps in Young's modulus. The results of various numerical experiments show that the proposed AMG method is much more robust and efficient than a classical AMG solver that is applied directly to the high‐order systems alone. Moreover, we present the corresponding theoretical analysis for the convergence of the proposed AMG algorithms. These theoretical results are also confirmed by some numerical tests. Copyright © 2008 John Wiley & Sons, Ltd. |
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| AbstractList | In this paper, we will discuss the geometric‐based algebraic multigrid (AMG) method for two‐dimensional linear elasticity problems discretized using quadratic and cubic elements. First, a two‐level method is proposed by analyzing the relationship between the linear finite element space and higher‐order finite element space. And then a geometric‐based AMG method is obtained with the existing solver used as a solver on the first coarse level. The resulting AMG method is applied to some typical elasticity problems including the plane strain problem with jumps in Young's modulus. The results of various numerical experiments show that the proposed AMG method is much more robust and efficient than a classical AMG solver that is applied directly to the high‐order systems alone. Moreover, we present the corresponding theoretical analysis for the convergence of the proposed AMG algorithms. These theoretical results are also confirmed by some numerical tests. Copyright © 2008 John Wiley & Sons, Ltd. In this paper, we will discuss the geometric-based algebraic multigrid (AMG) method for two-dimensional linear elasticity problems discretized using quadratic and cubic elements. First, a two-level method is proposed by analyzing the relationship between the linear finite element space and higher-order finite element space. And then a geometric-based AMG method is obtained with the existing solver used as a solver on the first coarse level. The resulting AMG method is applied to some typical elasticity problems including the plane strain problem with jumps in Young's modulus. The results of various numerical experiments show that the proposed AMG method is much more robust and efficient than a classical AMG solver that is applied directly to the high-order systems alone. Moreover, we present the corresponding theoretical analysis for the convergence of the proposed AMG algorithms. These theoretical results are also confirmed by some numerical tests. |
| Author | Zhao, Tuyan Shu, Shi Xiao, Yingxiong |
| Author_xml | – sequence: 1 givenname: Yingxiong surname: Xiao fullname: Xiao, Yingxiong organization: Civil Engineering and Mechanics College, Xiangtan University, Xiangtan 411105, People's Republic of China – sequence: 2 givenname: Shi surname: Shu fullname: Shu, Shi email: shushi@xtu.edu.cn organization: School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, People's Republic of China – sequence: 3 givenname: Tuyan surname: Zhao fullname: Zhao, Tuyan organization: Department of Mathematics and Physics, Hunan Institute of Engineering, Xiangtan 411104, People's Republic of China |
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| References | Shu S, Sun D, Xu J. An algebraic multigrids for higher order finite element discretizations. Computing 2006; 77:347-377. Xiao YX, Zhang P, Shu S. An algebraic multigrid method with interpolation reproducing rigid body modes for semi-definite problems in two-dimensional linear elasticity. Journal of Computational and Applied Mathematics 2007; 200(2):637-652. Trottenberg U, Osterlee CW, Schuler A. Multigrid Methods. Academic Press: San Diego, CA, 2001. Deville MO, Mund EH. Finite-element preconditioning for pseudospectral solutions of elliptic problems. SIAM Journal on Scientific and Statistical Computing 1990; 11:311-342. Xiao YX, Zhang P, Shu S. Algebraic multigrid method for a multiscale elastic model of thin epitaxial films. Natural Sciences Journal of Xiangtan University 2003; 25:75-82. Brandt A. Algebraic multigrid theory: the symmetric case. Applied Mathematics and Computation 1986; 19:23-56. Beuchler S. Multigrid solver for the inner problem in domain decomposition methods for p-fem. SIAM Journal on Numerical Analysis 2002; 40:928-944. Griebel M, Oeltz D, Schweitzer MA. An algebraic multigrid method for linear elasticity. SIAM Journal on Scientific Computing 2003; 25:385-407. Xu J, Zikatanov L. The method of alternating projections and the method of subspace corrections on hilbert space. Journal of the American Mathematics Society 2002; 15:573-597. Mandel J, Brezina M, Vanek P. Energy optimization of algebraic multigrid bases. Computing 1999; 62:205-228. Xiao YX, Zhang P, Shu S. Algebraic multigrid methods for elastic structures with highly discontinuous coefficients. Mathematics and Computers in Simulation 2007; 76(4):249-262. Lottes JW, Fischer PF. Hybrid multigrid/Schwarz algorithms for the spectral element method. Journal of Scientific Computation 2005; 24:45-78. Shu S, Sun D. An algebraic multigrid method of the high order Lagrangian finite element equation in three dimensions. Chinese Journal of Numerical Mathematics and Applications 2005; 27:91-106. Xiao YX, Zhang P, Shu S, Yang Y. Algebraic multigrid method for three-dimensional elasticity problems on the partitions with equal algebraic structure on each lay. Engineering Mechanics 2005; 22(6):76-81. Xiao YX, Shu S, Zhang P, Mo Z, Xu J. A kind of semi-coarsing AMG method for two dimensional energy equations with three temperatures. Journal on Numerical Methods and Computer Applications 2003; 24(4):293-303. Xu J, Zikatanov L. On an energy minimizing basis for algebraic multigrid methods. Computing and Visualization in Science 2004; 7:121-127. Wan WL. Interface preserving coarsening multigrid for elliptic problems with highly discontinuous coefficients. Numerical Linear Algebra with Application 2000; 7:727-741. Brezina M, Cleary AJ, Falgout RD, Henson VE, Jones JE, Manteuffel TA, McCormick SF, Ruge JW. Algebraic multigrid based on element interpolation (AMGe). SIAM Journal on Scientific Computing 2000; 22:1570-1592. Chartier T, Falgout RD, Henson VE, Jones JE, Manteuffel T, McCormick SF, Ruge JW, Vassilevski PS. Spectral AMGe (ρAMGe). SIAM Journal on Scientific Computing 2003; 20:1-20. Heys JJ, Manteuffel TA, McCormick SF, Olson LN. Algebraic multigrid (AMG) for higher-order finite elements. Journal of Computational Physics 2005; 204:520-532. Chan TF, Xu J, Zikatanov L. An agglomeration multigrid method for unstructured grids. Contemporary Mathematics 1998; 218:67-81. Luo ZD. Basis of Theory and Application of Mixed Finite Element Methods. Education Press of SanDong Province: SanDong, 1996. 2007; 200 2002; 15 1990; 11 2001 2006; 77 2002; 40 2004; 7 2000; 22 2005; 204 2000; 7 1987 1998; 218 2003; 24 2003; 25 1996 1986; 19 1999; 62 2005; 27 2007; 76 2003; 20 2005; 22 1989 2005; 24 Xiao YX (e_1_2_1_9_2) 2003; 25 Trottenberg U (e_1_2_1_2_2) 2001 Luo ZD (e_1_2_1_24_2) 1996 Xiao YX (e_1_2_1_20_2) 2005; 22 e_1_2_1_6_2 Xiao YX (e_1_2_1_19_2) 2003; 24 e_1_2_1_7_2 e_1_2_1_4_2 e_1_2_1_5_2 Shu S (e_1_2_1_12_2) 2005; 27 e_1_2_1_11_2 e_1_2_1_22_2 e_1_2_1_3_2 e_1_2_1_23_2 e_1_2_1_10_2 e_1_2_1_21_2 e_1_2_1_15_2 e_1_2_1_16_2 e_1_2_1_13_2 e_1_2_1_14_2 e_1_2_1_25_2 e_1_2_1_8_2 e_1_2_1_17_2 e_1_2_1_18_2 |
| References_xml | – reference: Xiao YX, Zhang P, Shu S. Algebraic multigrid method for a multiscale elastic model of thin epitaxial films. Natural Sciences Journal of Xiangtan University 2003; 25:75-82. – reference: Lottes JW, Fischer PF. Hybrid multigrid/Schwarz algorithms for the spectral element method. Journal of Scientific Computation 2005; 24:45-78. – reference: Beuchler S. Multigrid solver for the inner problem in domain decomposition methods for p-fem. SIAM Journal on Numerical Analysis 2002; 40:928-944. – reference: Griebel M, Oeltz D, Schweitzer MA. An algebraic multigrid method for linear elasticity. SIAM Journal on Scientific Computing 2003; 25:385-407. – reference: Trottenberg U, Osterlee CW, Schuler A. Multigrid Methods. Academic Press: San Diego, CA, 2001. – reference: Shu S, Sun D, Xu J. An algebraic multigrids for higher order finite element discretizations. Computing 2006; 77:347-377. – reference: Xiao YX, Zhang P, Shu S. An algebraic multigrid method with interpolation reproducing rigid body modes for semi-definite problems in two-dimensional linear elasticity. Journal of Computational and Applied Mathematics 2007; 200(2):637-652. – reference: Heys JJ, Manteuffel TA, McCormick SF, Olson LN. Algebraic multigrid (AMG) for higher-order finite elements. Journal of Computational Physics 2005; 204:520-532. – reference: Brandt A. Algebraic multigrid theory: the symmetric case. Applied Mathematics and Computation 1986; 19:23-56. – reference: Xiao YX, Zhang P, Shu S. Algebraic multigrid methods for elastic structures with highly discontinuous coefficients. Mathematics and Computers in Simulation 2007; 76(4):249-262. – reference: Chartier T, Falgout RD, Henson VE, Jones JE, Manteuffel T, McCormick SF, Ruge JW, Vassilevski PS. Spectral AMGe (ρAMGe). SIAM Journal on Scientific Computing 2003; 20:1-20. – reference: Xiao YX, Zhang P, Shu S, Yang Y. Algebraic multigrid method for three-dimensional elasticity problems on the partitions with equal algebraic structure on each lay. Engineering Mechanics 2005; 22(6):76-81. – reference: Mandel J, Brezina M, Vanek P. Energy optimization of algebraic multigrid bases. Computing 1999; 62:205-228. – reference: Chan TF, Xu J, Zikatanov L. An agglomeration multigrid method for unstructured grids. Contemporary Mathematics 1998; 218:67-81. – reference: Xu J, Zikatanov L. On an energy minimizing basis for algebraic multigrid methods. Computing and Visualization in Science 2004; 7:121-127. – reference: Brezina M, Cleary AJ, Falgout RD, Henson VE, Jones JE, Manteuffel TA, McCormick SF, Ruge JW. Algebraic multigrid based on element interpolation (AMGe). SIAM Journal on Scientific Computing 2000; 22:1570-1592. – reference: Deville MO, Mund EH. Finite-element preconditioning for pseudospectral solutions of elliptic problems. SIAM Journal on Scientific and Statistical Computing 1990; 11:311-342. – reference: Luo ZD. Basis of Theory and Application of Mixed Finite Element Methods. Education Press of SanDong Province: SanDong, 1996. – reference: Xu J, Zikatanov L. The method of alternating projections and the method of subspace corrections on hilbert space. Journal of the American Mathematics Society 2002; 15:573-597. – reference: Xiao YX, Shu S, Zhang P, Mo Z, Xu J. A kind of semi-coarsing AMG method for two dimensional energy equations with three temperatures. Journal on Numerical Methods and Computer Applications 2003; 24(4):293-303. – reference: Shu S, Sun D. An algebraic multigrid method of the high order Lagrangian finite element equation in three dimensions. Chinese Journal of Numerical Mathematics and Applications 2005; 27:91-106. – reference: Wan WL. Interface preserving coarsening multigrid for elliptic problems with highly discontinuous coefficients. 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order finite element discretizations publication-title: Computing – volume: 11 start-page: 311 year: 1990 end-page: 342 article-title: Finite‐element preconditioning for pseudospectral solutions of elliptic problems publication-title: SIAM Journal on Scientific and Statistical Computing – volume: 24 start-page: 293 issue: 4 year: 2003 end-page: 303 article-title: A kind of semi‐coarsing AMG method for two dimensional energy equations with three temperatures publication-title: Journal on Numerical Methods and Computer Applications – volume: 22 start-page: 76 issue: 6 year: 2005 end-page: 81 article-title: Algebraic multigrid method for three‐dimensional elasticity problems on the partitions with equal algebraic structure on each lay publication-title: Engineering Mechanics – volume: 200 start-page: 637 issue: 2 year: 2007 end-page: 652 article-title: An algebraic multigrid method with interpolation reproducing rigid body modes for semi‐definite problems in two‐dimensional linear elasticity publication-title: Journal of Computational and Applied Mathematics – volume: 22 start-page: 1570 year: 2000 end-page: 1592 article-title: Algebraic multigrid based on element interpolation (AMGe) publication-title: SIAM Journal on Scientific Computing – volume-title: Multigrid Methods year: 2001 ident: e_1_2_1_2_2 – ident: e_1_2_1_3_2 doi: 10.1016/0096-3003(86)90095-0 – ident: e_1_2_1_7_2 doi: 10.1137/S1064827502407810 – volume: 27 start-page: 91 year: 2005 ident: e_1_2_1_12_2 article-title: An algebraic multigrid method of the high order Lagrangian finite element equation in three dimensions publication-title: Chinese Journal of Numerical Mathematics and Applications – ident: e_1_2_1_15_2 doi: 10.1007/s10915-004-4787-3 – ident: e_1_2_1_5_2 doi: 10.1007/s006070050022 – ident: e_1_2_1_14_2 doi: 10.1007/3-540-51048-6_61 – volume: 22 start-page: 76 issue: 6 year: 2005 ident: e_1_2_1_20_2 article-title: Algebraic multigrid method for three‐dimensional elasticity problems on the partitions with equal algebraic structure on each lay publication-title: Engineering Mechanics – volume: 25 start-page: 75 year: 2003 ident: e_1_2_1_9_2 article-title: Algebraic multigrid method for a multiscale elastic model of thin epitaxial films publication-title: Natural Sciences Journal of Xiangtan University – ident: e_1_2_1_10_2 doi: 10.1090/conm/218/03002 – ident: e_1_2_1_22_2 doi: 10.1016/j.matcom.2006.10.025 – ident: e_1_2_1_16_2 doi: 10.1137/0911019 – volume-title: Basis of Theory and Application of Mixed Finite Element Methods year: 1996 ident: e_1_2_1_24_2 – volume: 24 start-page: 293 issue: 4 year: 2003 ident: e_1_2_1_19_2 article-title: A kind of semi‐coarsing AMG method for two dimensional energy equations with three temperatures publication-title: Journal on Numerical Methods and Computer Applications – ident: e_1_2_1_25_2 doi: 10.1002/1099-1506(200010/12)7:7/8<727::AID-NLA221>3.0.CO;2-H – ident: e_1_2_1_6_2 doi: 10.1007/s00791-004-0147-y – ident: e_1_2_1_23_2 doi: 10.1016/j.jcp.2004.10.021 – ident: e_1_2_1_13_2 doi: 10.1007/s00607-006-0162-6 – ident: e_1_2_1_17_2 doi: 10.1137/S0036142901393851 – ident: e_1_2_1_18_2 doi: 10.1090/S0894-0347-02-00398-3 – ident: e_1_2_1_4_2 doi: 10.1137/1.9781611971057.ch4 – ident: e_1_2_1_8_2 doi: 10.1137/S1064827598344303 – ident: e_1_2_1_11_2 doi: 10.1137/S106482750139892X – ident: e_1_2_1_21_2 doi: 10.1016/j.cam.2006.01.021 |
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| Title | A geometric-based algebraic multigrid method for higher-order finite element equations in two-dimensional linear elasticity |
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