## An Algorithm for Coarsening Unstructured Meshes (1996)

Venue: | Numer. Math |

Citations: | 49 - 5 self |

### BibTeX

@ARTICLE{Bank96analgorithm,

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

title = {An Algorithm for Coarsening Unstructured Meshes},

journal = {Numer. Math},

year = {1996},

volume = {73},

pages = {1--36}

}

### Years of Citing Articles

### OpenURL

### Abstract

. We develop and analyze a procedure for creating a hierarchical basis of continuous piecewise linear polynomials on an arbitrary, unstructured, nonuniform triangular mesh. Using these hierarchical basis functions, we are able to define and analyze corresponding iterative methods for solving the linear systems arising from finite element discretizations of elliptic partial differential equations. We show that such iterative methods perform as well as those developed for the usual case of structured, locally refined meshes. In particular, we show that the generalized condition numbers for such iterative methods are of order J 2 , where J is the number of hierarchical basis levels. Key words. Finite element, hierarchical basis, multigrid, unstructured mesh. AMS subject classifications. 65F10, 65N20 1. Introduction. Iterative methods using the hierarchical basis decomposition have proved to be among the most robust for solving broad classes of elliptic partial differential equations, ...

### Citations

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Citation Context ...rising 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 [27][28][4]=-=[17]-=-. Classical hierarchical basis and multigrid methods are defined in terms of an underlying refinement structure of a sequence of nested meshes. In many cases this is no disadvantage, but it limits the... |

534 |
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Citation Context ...we must allow nonconvex quadrilaterals (see Figure 2 F). We begin with some notation, definitions and descriptions which we need to describe the coarsening algorithm. See Rose [26] and George and Liu =-=[13]-=- for a complete discussion of the connection of graph theory and Gaussian elimination. To keep the notation as simple as possible, we will drop subscripts unless absolutely necessary. Let T be a graph... |

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Citation Context ...k D \Gamma1=2 ZD \Gamma1=2 k ` 2=s1 p 2sC: Combining these estimates, we see thatsCJ 2 , just as in the basic block symmetric Gauss-Seidel iteration. We next formally define algorithms similar to BPX =-=[10]-=- and regular multigrid algorithms based on our decomposition. We will use the ideas of Griebel [15] [14], who characterizes these methods as particular iterations applied to an enlarged, semidefinite ... |

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Citation Context ...condition number for these schemes grows as J 2 , where J is the number of hierarchical basis levels. This is essentially the same result as for the classical hierarchical basis schemes. [1] [4] [28] =-=[29]-=- [30] [23] [24] [8]. We can also define regular multigrid and BPX like methods based on the hierarchical basis decompositions. These methods are based on the observation of Griebel [15] [14] and Grieb... |

122 |
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Citation Context ...ed to be among the most robust 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 [27][28][4][17]. Classical hierarchical basis and multigrid methods are defined in ... |

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Citation Context ...s 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 [27][28]=-=[4]-=-[17]. Classical hierarchical basis and multigrid methods are defined in terms of an underlying refinement structure of a sequence of nested meshes. In many cases this is no disadvantage, but it limits... |

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Citation Context ... meshes, however, since we must allow nonconvex quadrilaterals (see Figure 2 F). We begin with some notation, definitions and descriptions which we need to describe the coarsening algorithm. See Rose =-=[26]-=- and George and Liu [13] for a complete discussion of the connection of graph theory and Gaussian elimination. To keep the notation as simple as possible, we will drop subscripts unless absolutely nec... |

94 |
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Citation Context ...ows us to extend the hierarchical basis and other related iterative methods in a natural way to such meshes. Some work on multigrid methods on non-nested meshes is reported in Bramble, Pasciak and Xu =-=[9]-=-, Xu [27], Zhang [31], Chan and Smith [11], Mavriplis [21], Kornhuber [20], Hoppe and Kornhuber [19], and Bank and Xu [6] [7]. The practical algorithm for unrefining an arbitrary mesh is developed in ... |

87 |
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Citation Context ...ape regularity concerns influence the selection of the subset F k\Gamma1 in Step 1. While there are several good strategies for controlling shape regularity during the refinement process [5] [2] [22] =-=[25]-=-, their details do not concern us here. Third, we are assuming an edge based model for refinement; it is possible to also have an element based model, where an element chosen for refinement is divided... |

83 |
Theory of Multilevel Methods
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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 =-=[27]-=-[28][4][17]. Classical hierarchical basis and multigrid methods are defined in terms of an underlying refinement structure of a sequence of nested meshes. In many cases this is no disadvantage, but it... |

78 |
A comparison of adaptive refinement techniques for elliptic problems
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Citation Context ...d. Shape regularity concerns influence the selection of the subset F k\Gamma1 in Step 1. While there are several good strategies for controlling shape regularity during the refinement process [5] [2] =-=[22]-=- [25], their details do not concern us here. Third, we are assuming an edge based model for refinement; it is possible to also have an element based model, where an element chosen for refinement is di... |

40 | Multilevel algorithms considered as iterative methods on semidefinite systems
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Citation Context ... [4] [28] [29] [30] [23] [24] [8]. We can also define regular multigrid and BPX like methods based on the hierarchical basis decompositions. These methods are based on the observation of Griebel [15] =-=[14]-=- and Griebel and Oswald [16], who show that such methods can be interpreted as standard block iterations applied to a larger, singular system of equations. Finally, in Section 6, we give some numerica... |

37 | Analysis of a two-level scheme for solving finite element equations, tech
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Citation Context ...` \Gamma kjjjujjj ~ t : The result (16) follows by summing over t 2 T k . The next lemma shows that a bound on the interpolant as in (16) in essentially equivalent to a strengthened Cauchy inequality =-=[3]-=- [12]. Lemma 4.4. Suppose M = V \Phi W, and let I denote the interpolation operator defined as follows: if u = v + w 2 M, v 2 V, and w 2 W, then I(u) = v. Then jjjI(u)jjj DsCjjjujjj D (17) if and only... |

37 | Domain decomposition and multigrid algorithms for elliptic problems on unstructured meshes
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Citation Context ...nd other related iterative methods in a natural way to such meshes. Some work on multigrid methods on non-nested meshes is reported in Bramble, Pasciak and Xu [9], Xu [27], Zhang [31], Chan and Smith =-=[11]-=-, Mavriplis [21], Kornhuber [20], Hoppe and Kornhuber [19], and Bank and Xu [6] [7]. The practical algorithm for unrefining an arbitrary mesh is developed in Sections 2 and 3. Our basic idea is to for... |

30 | Adaptive multilevel methods for obstacle problems
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Citation Context ...h meshes. Some work on multigrid methods on non-nested meshes is reported in Bramble, Pasciak and Xu [9], Xu [27], Zhang [31], Chan and Smith [11], Mavriplis [21], Kornhuber [20], Hoppe and Kornhuber =-=[19]-=-, and Bank and Xu [6] [7]. The practical algorithm for unrefining an arbitrary mesh is developed in Sections 2 and 3. Our basic idea is to force an arbitrary unstructured mesh into the mold of a possi... |

27 |
Multilevel iterative techniques
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(Show Context)
Citation Context ... hierarchical basis and other related iterative methods in a natural way to such meshes. Some work on multigrid methods on non-nested meshes is reported in Bramble, Pasciak and Xu [9], Xu [27], Zhang =-=[31]-=-, Chan and Smith [11], Mavriplis [21], Kornhuber [20], Hoppe and Kornhuber [19], and Bank and Xu [6] [7]. The practical algorithm for unrefining an arbitrary mesh is developed in Sections 2 and 3. Our... |

26 | The hierarchical basis multigrid method and incomplete LU decomposition
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(Show Context)
Citation Context ... multigrid methods on non-nested meshes is reported in Bramble, Pasciak and Xu [9], Xu [27], Zhang [31], Chan and Smith [11], Mavriplis [21], Kornhuber [20], Hoppe and Kornhuber [19], and Bank and Xu =-=[6]-=- [7]. The practical algorithm for unrefining an arbitrary mesh is developed in Sections 2 and 3. Our basic idea is to force an arbitrary unstructured mesh into the mold of a possibly nonuniform, local... |

22 | Multigrid techniques for unstructured meshes
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Citation Context ... iterative methods in a natural way to such meshes. Some work on multigrid methods on non-nested meshes is reported in Bramble, Pasciak and Xu [9], Xu [27], Zhang [31], Chan and Smith [11], Mavriplis =-=[21]-=-, Kornhuber [20], Hoppe and Kornhuber [19], and Bank and Xu [6] [7]. The practical algorithm for unrefining an arbitrary mesh is developed in Sections 2 and 3. Our basic idea is to force an arbitrary ... |

22 |
new convergence proofs for multigrid methods. Acta Numerica
- Old
- 1993
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Citation Context ...tion number for these schemes grows as J 2 , where J is the number of hierarchical basis levels. This is essentially the same result as for the classical hierarchical basis schemes. [1] [4] [28] [29] =-=[30]-=- [23] [24] [8]. We can also define regular multigrid and BPX like methods based on the hierarchical basis decompositions. These methods are based on the observation of Griebel [15] [14] and Griebel an... |

20 | The role of the strengthened Cauchyâ€“Buniakowskii inequality in multilevel methods
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Citation Context ...amma kjjjujjj ~ t : The result (16) follows by summing over t 2 T k . The next lemma shows that a bound on the interpolant as in (16) in essentially equivalent to a strengthened Cauchy inequality [3] =-=[12]-=-. Lemma 4.4. Suppose M = V \Phi W, and let I denote the interpolation operator defined as follows: if u = v + w 2 M, v 2 V, and w 2 W, then I(u) = v. Then jjjI(u)jjj DsCjjjujjj D (17) if and only if j... |

10 |
Remarks on the abstract theory of additive and multiplicative Schwarz methods
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(Show Context)
Citation Context ...] [8]. We can also define regular multigrid and BPX like methods based on the hierarchical basis decompositions. These methods are based on the observation of Griebel [15] [14] and Griebel and Oswald =-=[16]-=-, who show that such methods can be interpreted as standard block iterations applied to a larger, singular system of equations. Finally, in Section 6, we give some numerical illustrations and provide ... |

8 |
A Software Package for Solving Elliptic Partial Differential Equations
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Citation Context ...to be among the most robust 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 [27][28][4][17]. Classical hierarchical basis and multigrid methods are defined in ter... |

7 |
Monotone multigrid methods for variational inequalities
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(Show Context)
Citation Context ...ds in a natural way to such meshes. Some work on multigrid methods on non-nested meshes is reported in Bramble, Pasciak and Xu [9], Xu [27], Zhang [31], Chan and Smith [11], Mavriplis [21], Kornhuber =-=[20]-=-, Hoppe and Kornhuber [19], and Bank and Xu [6] [7]. The practical algorithm for unrefining an arbitrary mesh is developed in Sections 2 and 3. Our basic idea is to force an arbitrary unstructured mes... |

6 |
Finite Element Approximation Theory and Applications, Teubner Skripten zur Numerik
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Citation Context ...r for these schemes grows as J 2 , where J is the number of hierarchical basis levels. This is essentially the same result as for the classical hierarchical basis schemes. [1] [4] [28] [29] [30] [23] =-=[24]-=- [8]. We can also define regular multigrid and BPX like methods based on the hierarchical basis decompositions. These methods are based on the observation of Griebel [15] [14] and Griebel and Oswald [... |

2 |
Hierarchical preconditioners for elliptic partial differential equations
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(Show Context)
Citation Context ... generalized condition number for these schemes grows as J 2 , where J is the number of hierarchical basis levels. This is essentially the same result as for the classical hierarchical basis schemes. =-=[1]-=- [4] [28] [29] [30] [23] [24] [8]. We can also define regular multigrid and BPX like methods based on the hierarchical basis decompositions. These methods are based on the observation of Griebel [15] ... |

1 |
A basic norm equivalence for the theory of multigrid methods
- Bornemann, Yserentant
- 1993
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Citation Context ... these schemes grows as J 2 , where J is the number of hierarchical basis levels. This is essentially the same result as for the classical hierarchical basis schemes. [1] [4] [28] [29] [30] [23] [24] =-=[8]-=-. We can also define regular multigrid and BPX like methods based on the hierarchical basis decompositions. These methods are based on the observation of Griebel [15] [14] and Griebel and Oswald [16],... |

1 |
als Interationsverfahren Erzeugendensystemen
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Citation Context .... [1] [4] [28] [29] [30] [23] [24] [8]. We can also define regular multigrid and BPX like methods based on the hierarchical basis decompositions. These methods are based on the observation of Griebel =-=[15]-=- [14] and Griebel and Oswald [16], who show that such methods can be interpreted as standard block iterations applied to a larger, singular system of equations. Finally, in Section 6, we give some num... |

1 |
On discrete norm estimates related to multigrid preconditioners in the finite element method
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- 1991
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Citation Context ...number for these schemes grows as J 2 , where J is the number of hierarchical basis levels. This is essentially the same result as for the classical hierarchical basis schemes. [1] [4] [28] [29] [30] =-=[23]-=- [24] [8]. We can also define regular multigrid and BPX like methods based on the hierarchical basis decompositions. These methods are based on the observation of Griebel [15] [14] and Griebel and Osw... |