## The Hierarchical Basis Multigrid Method And Incomplete LU Decomposition (1994)

Venue: | In Seventh International Symposium on Domain Decomposition Methods for Partial Differential Equations |

Citations: | 26 - 7 self |

### BibTeX

@INPROCEEDINGS{Bank94thehierarchical,

author = {Randolph E. Bank and jinchao Xu},

title = {The Hierarchical Basis Multigrid Method And Incomplete LU Decomposition},

booktitle = {In Seventh International Symposium on Domain Decomposition Methods for Partial Differential Equations},

year = {1994},

pages = {163--173},

publisher = {AMS}

}

### OpenURL

### Abstract

. A new multigrid or incomplete LU technique is developed in this paper for solving large sparse algebraic systems from discretizing partial differential equations. By exploring some deep connection between the hierarchical basis method and incomplete LU decomposition, the resulting algorithm can be effectively applied to problems discretized on completelyunstructured grids. Numerical experiments demonstrating the efficiency of the method are also reported. Key words. Finite element, hierarchical basis, multigrid, incomplete LU . AMS(MOS) subject classifications. 65F10, 65N20 1. Introduction. In this work, we explore the connection between the methods of sparse Gaussian elimination [8][13], incomplete LU (ILU) decomposition [9][10] and the hierarchical basis multigrid (HBMG) [16][4]. Hierarchical basis methods have proved to be one of the more robust classes of methods for solving broad classes of elliptic partial differential equations, especially the large systems arising in conju...

### Citations

761 |
Multi-grid methods and applications
- Hackbusch
- 1985
(Show Context)
Citation Context .... AMS(MOS) subject classifications. 65F10, 65N20 1. Introduction. In this work, we explore the connection between the methods of sparse Gaussian elimination [8][13], incomplete LU (ILU) decomposition =-=[9]-=-[10] and the hierarchical basis multigrid (HBMG) [16][4]. Hierarchical basis methods have proved to be one of the more robust classes of methods for solving broad classes of elliptic partial different... |

500 |
Computer Solution of Large Sparse Positive Definite Systems
- George, Liu
- 1981
(Show Context)
Citation Context ...erarchical basis, multigrid, incomplete LU . AMS(MOS) subject classifications. 65F10, 65N20 1. Introduction. In this work, we explore the connection between the methods of sparse Gaussian elimination =-=[8]-=-[13], incomplete LU (ILU) decomposition [9][10] and the hierarchical basis multigrid (HBMG) [16][4]. Hierarchical basis methods have proved to be one of the more robust classes of methods for solving ... |

226 | On the multi-level-splitting of the finite element spaces
- Yserentant
- 1986
(Show Context)
Citation Context ...Introduction. In this work, we explore the connection between the methods of sparse Gaussian elimination [8][13], incomplete LU (ILU) decomposition [9][10] and the hierarchical basis multigrid (HBMG) =-=[16]-=-[4]. Hierarchical basis methods have proved to be one of the more robust classes of methods for solving broad classes of elliptic partial differential equations, especially the large systems arising i... |

117 |
Refinement algorithms and data structures for regular local mesh refinement
- Bank, Sherman, et al.
- 1983
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Citation Context ...more robust classes of methods for solving broad classes of elliptic partial differential equations, especially the large systems arising in conjunction with adaptive local mesh refinement techniques =-=[5]-=-[2]; they have been shown to be strongly connected to space decomposition methods and to classical multigrid methods [14][15][4][9]. As with typical multigrid methods, classical hierarchical basis met... |

104 | The Hierarchical Basis Multigrid Method
- BANK, DUPONT, et al.
- 1988
(Show Context)
Citation Context ...oduction. In this work, we explore the connection between the methods of sparse Gaussian elimination [8][13], incomplete LU (ILU) decomposition [9][10] and the hierarchical basis multigrid (HBMG) [16]=-=[4]-=-. Hierarchical basis methods have proved to be one of the more robust classes of methods for solving broad classes of elliptic partial differential equations, especially the large systems arising in c... |

103 |
A graph-theoretic study of the numerical solution of sparse positive definite systems of linear equations, Graph Theory and Computing
- Rose
- 1972
(Show Context)
Citation Context ...rchical basis, multigrid, incomplete LU . AMS(MOS) subject classifications. 65F10, 65N20 1. Introduction. In this work, we explore the connection between the methods of sparse Gaussian elimination [8]=-=[13]-=-, incomplete LU (ILU) decomposition [9][10] and the hierarchical basis multigrid (HBMG) [16][4]. Hierarchical basis methods have proved to be one of the more robust classes of methods for solving broa... |

84 |
The analysis of multigrid algorithms with nonnested spaces or noninherited quadratic forms
- BRAMBLE, PASCIAK, et al.
- 1991
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Citation Context ...tion of hierarchical bases to such meshes, allowing HBMG and other hierarchical basis methods to be applied. Some work on multigrid methods on non-nested meshes is reported in Bramble, Pasciak and Xu =-=[6]-=-, Xu [14], and Zhang [17]. In Section 2, we develop a simple graph elimination model for classical hierarchical basis methods on sequences of nested meshes. This elimination model can be interpreted a... |

74 |
Theory of multilevel methods
- Xu
- 1989
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Citation Context ...e systems arising in conjunction with adaptive local mesh refinement techniques [5][2]; they have been shown to be strongly connected to space decomposition methods and to classical multigrid methods =-=[14]-=-[15][4][9]. As with typical multigrid methods, classical hierarchical basis methods are usually defined in terms of an underlying refinement structure of a sequence of nested meshes. In many cases thi... |

35 | Analysis of a two level scheme for solving finite element equations
- Bank, Dupont
- 1980
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Citation Context ...section. The matrix RA f + C t is not zero as in the case of Gaussian elimination; indeed it is less sparse than C t . However, the matrix is small in some sense; the usual Cauchy inequality estimate =-=[3]-=- [1] [7] written in this notation is: jx t (RA f + C t )yjsfl(x tsA c x) 1=2 (y t A f y) 1=2 ; where fl ! 1 is the constant in the strengthened Cauchy inequality. It is worth commenting that in implem... |

27 |
Multi-level iterative techniques
- Zhang
- 1988
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Citation Context ...es to such meshes, allowing HBMG and other hierarchical basis methods to be applied. Some work on multigrid methods on non-nested meshes is reported in Bramble, Pasciak and Xu [6], Xu [14], and Zhang =-=[17]-=-. In Section 2, we develop a simple graph elimination model for classical hierarchical basis methods on sequences of nested meshes. This elimination model can be interpreted as a particular ILU decomp... |

26 | Adaptive multilevel methods for obstacle problems
- Hoppe, Kornhuber
- 1994
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Citation Context ... long as the complicated basis functions have that property, one should see the expected convergence rates. The use of modified hierarchical basis functions appears in the work of Hoppe and Kornhuber =-=[11]-=- and Kornhuber [12] in connection with the the solution of obstacle problems. We now consider the construction of R for the two heuristic algorithms for un7 structured meshes described in the last sec... |

19 | The role of the strengthened Cauchy-Buniakowskii-Schwarz inequality in multilevel methods
- Eijkhout, Vassilevski
- 1991
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Citation Context ... The matrix RA f + C t is not zero as in the case of Gaussian elimination; indeed it is less sparse than C t . However, the matrix is small in some sense; the usual Cauchy inequality estimate [3] [1] =-=[7]-=- written in this notation is: jx t (RA f + C t )yjsfl(x tsA c x) 1=2 (y t A f y) 1=2 ; where fl ! 1 is the constant in the strengthened Cauchy inequality. It is worth commenting that in implementation... |

8 |
A Software Package for Solving Elliptic Partial Differential Equations
- PLTMG
- 1994
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Citation Context ...e robust classes of methods for solving broad classes of elliptic partial differential equations, especially the large systems arising in conjunction with adaptive local mesh refinement techniques [5]=-=[2]-=-; they have been shown to be strongly connected to space decomposition methods and to classical multigrid methods [14][15][4][9]. As with typical multigrid methods, classical hierarchical basis method... |

6 |
Monotone multigrid methods for variational inequalities
- Kornhuber
- 1996
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Citation Context ...cated basis functions have that property, one should see the expected convergence rates. The use of modified hierarchical basis functions appears in the work of Hoppe and Kornhuber [11] and Kornhuber =-=[12]-=- in connection with the the solution of obstacle problems. We now consider the construction of R for the two heuristic algorithms for un7 structured meshes described in the last section. In both cases... |

2 |
Hierarchical preconditioners for elliptic partial differential equations
- Bank
- 1994
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Citation Context ..., but at the cost of having fairly dense matrices R andsA c . In the case of classical HBMG, the matrix R t is sparse, and contains information about changing from the nodal to hierarchical basis [4],=-=[1]-=-. Each row of R t is zero except for two entries which are equal to 1=2. For the row corresponding to vertex v i 2 X f , the two nonzero entries are in the columns corresponding to v j 2 X c and v k 2... |