## Well-Spaced Points for Numerical Methods (1997)

Citations: | 44 - 2 self |

### BibTeX

@TECHREPORT{Talmor97well-spacedpoints,

author = {Dafna Talmor and Guy Blelloch and Alan M. Frieze and Noel J. Walkington and Shang-hua Teng},

title = {Well-Spaced Points for Numerical Methods},

institution = {},

year = {1997}

}

### OpenURL

### Abstract

mesh generation, mesh coarsening, multigrid Abstract A numerical method for the solution of a partial differential equation (PDE) requires the following steps: (1) discretizing the domain (mesh generation); (2) using an approximation method and the mesh to transform the problem into a linear system; (3) solving the linear system. The approximation error and convergence of the numerical method depend on the geometric quality of the mesh, which in turn depends on the size and shape of its elements. For example, the shape quality of a triangular mesh is measured by its element's aspect ratio. In this work, we shift the focus to the geometric properties of the nodes, rather than the elements, of well shaped meshes. We introduce the concept of well-spaced points and their spacing functions, and show that these enable the development of simple and efficient algorithms for the different stages of the numerical solution of PDEs. We first apply well-spaced point sets and their accompanying technology to mesh coarsening, a crucial step in the multigrid solution of a PDE. A good aspect-ratio coarsening sequence of an unstructured mesh M0 is a sequence of good aspect-ratio meshes M1; : : : ; Mk such that Mi is an approximation of Mi\Gamma 1 containing fewer nodes and elements. We present a new approach to coarsening that guarantees the sequence is also of optimal size and width up to a constant factor-- the first coarsening method that provides these guarantees. We also present experimental results, based on an implementation of our approach, that substantiate the theoretical claims.

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Citation Context ...ecide if a polyhedron is tetrahedralizable, or how many Steiner points are necessary to tetrahedralize it. Some domains may require O(n2) Steiner points to be added before they can be tetrahedralized =-=[18]-=-. The closer the polyhedron is to a convex polyhedron, the fewer Steiner points are necessary. The number of reflex angles in a polyhedron is a measure of how close to being convex the polyhedron is. ... |

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Citation Context ...elements in the generated mesh. 49sCHAPTER 4. GENERATING 3D MESHES FROM WELL-SPACED POINTS Figure 4.1: A domain that can not be tetrahedralized. This example is based on an example due to Sch"onhardt =-=[66]-=-. Starting from a cube, twist the top square clockwise very slightly. As a result, the dashed lines appear as edges with a reflex angle. This twisted cube can not be tetrahedralized because the standa... |

48 | An Algorithm for Coarsening Unstructured Meshes
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Citation Context ...rser elements. The theory of multigrid for nested elements is better understood. However, this method is not viable for a black box approach. Of interest is the hybrid method suggested by Bank and Xu =-=[5]-=-. They try to implement a derefinement method without the intermediate information. The input fine mesh is assumed to have been constructed by a nested elements refinement process, which they aim to r... |

48 | mesh generation
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Citation Context ...rary refinings. The implication of the reversal of the temporary refining is that the method does not yield a nested coarsening sequence. refinement, see [62, 4, 41, 53] and the two excellent surveys =-=[10, 8]-=-. These papers present a variety of refinement techniques, such as longest edge bisection [70, 62, 63], newest vertex bisection [53, 54], vertex ordered simplicial bisection [44] and regular refinemen... |

46 |
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Citation Context ...to form a new larger cell. As in the MIS-based schemes, the order by which the volumes are traversed can be prioritized to proceed from the boundary cells inward, or in an advanced front fashion. See =-=[72, 48, 46, 40]-=-. 74s5.2. PREVIOUS WORK Figure 5.3: Repeated applications of MIS can degrade the aspect ratio. 5.2.2 Repeated degradation: problems of topological coarsening MIS on the 1D skeleton is very successful ... |

44 | Direct discretization of planar div-curl problems
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Citation Context ...n; however, this is not the case in three dimensions. 15sCHAPTER 1. INTRODUCTION p i p j Vi Aij h ij Figure 1.8: The Voronoi diagram of pi, the Delaunay edge hij, and the Voronoi face Aij. Nicolaides =-=[55]-=- established error estimates for the control volume method. For quantities defined on the edges of the Delaunay mesh Nicolaides defines the inner product (:; :)W (U; V )W = X (i;j) j AijjhijUijVij; an... |

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Citation Context ... shape and size. Examples for such algorithms can be found in the work of Bern, Eppstein and Gilbert [9], Chew [21], Ruppert [64], Mitchell and Vavasis [52], Miller, Talmor, Teng, Walkington and Wang =-=[50]-=- and Shewchuk [67]. These algorithms take as input a piecewise linear description of the domain, and partition it into triangles in two dimensions, or tetrahedra in three dimensions. The elements' sha... |

37 |
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Citation Context ...und on the dilation of the Euclidean Delaunay triangulation. They showed it is at most (1 + p5)\Pi =2 ss 5:083. This bound has later been improved to 2\Pi =(3 cos(\Pi =6)) ss 2:418 by Keil and Gutwin =-=[39]-=-. Eppstein [28] in his survey "Spanning Trees and Spanners" describes this and other related work in the larger context of spanners, graphs of low dilation. The dilation of the Euclidean Delaunay tria... |

36 |
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Citation Context ... not yield a nested coarsening sequence. refinement, see [62, 4, 41, 53] and the two excellent surveys [10, 8]. These papers present a variety of refinement techniques, such as longest edge bisection =-=[70, 62, 63]-=-, newest vertex bisection [53, 54], vertex ordered simplicial bisection [44] and regular refinement [4, 3]. If all intermediate meshes have been kept, mesh coarsening can be obtained by undoing the ch... |

35 |
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Citation Context ...tween any two intersecting polytopes, when one is not contained in the other, is at least 90ffi. The notion of a piecewise linear system is similar to a convex decomposition of non-manifold polyhedra =-=[2]-=-, but is more general. For example, a box with a segment in the interior can be described as a piecewise linear system with no need to further decompose it. 4.3 Algorithm "packing" Definition 4.3.1 (p... |

35 |
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Citation Context ...yhedron is to a convex polyhedron, the fewer Steiner points are necessary. The number of reflex angles in a polyhedron is a measure of how close to being convex the polyhedron is. Chazelle and Palios =-=[19]-=- have shown that any simple polyhedron with n nodes and r reflex edges can be partitioned into O(n + r2) elements using O(n + r2) Steiner points. Provably good mesh generation: The problem of provably... |

34 | Domain decomposition and multigrid algorithms for elliptic problems on unstructured meshes, Electron
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(Show Context)
Citation Context ...behind applying multigrid to unstructured meshes is not as well understood as it is for the structured and nested case. However, it is an area of active research, with increasing theoretical progress =-=[11, 15, 17]-=-. In practice, multigrid methods on unstructured, non-nested mesh sequences are popular despite this current lack of full theoretical foundations [6, 57]. The effectiveness of a multi-level method tha... |

33 |
Output-sensitive construction of polytopes in four dimensions T.M. Chan and clipped Voronoi diagrams in three
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Citation Context ...lipping-based approach. The Delaunay tetrahedralization can be computed directly, using four dimensional convex hull algorithms, or incrementally using Watson's algorithm [73]. Chan, Snoeyink and Yap =-=[16]-=- gave an output-sensitive algorithm for Delaunay diagrams computations. Since the size of the Delaunay diagram varies from O(n) to O(n2), the complexity of their algorithm can be much smaller than the... |

33 |
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Citation Context ...us, this form of refining uses auxiliary temporary refinings. The implication of the reversal of the temporary refining is that the method does not yield a nested coarsening sequence. refinement, see =-=[62, 4, 41, 53]-=- and the two excellent surveys [10, 8]. These papers present a variety of refinement techniques, such as longest edge bisection [70, 62, 63], newest vertex bisection [53, 54], vertex ordered simplicia... |

27 | Additive Schwarz domain decomposition methods for elliptic problems on unstructured meshes
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(Show Context)
Citation Context ...behind applying multigrid to unstructured meshes is not as well understood as it is for the structured and nested case. However, it is an area of active research, with increasing theoretical progress =-=[11, 15, 17]-=-. In practice, multigrid methods on unstructured, non-nested mesh sequences are popular despite this current lack of full theoretical foundations [6, 57]. The effectiveness of a multi-level method tha... |

24 |
Structured Background Grids for Generation of Unstructured Grids by Advancing Front Method
- Pirzadeh
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(Show Context)
Citation Context ...pular way to represent explicit functions is using background grids, with function values at the grid nodes interpolated to interior nodes. The background grid can be unstructured [58, 35], Cartesian =-=[59]-=-, or quadtree based [45]. ffl In mesh size analysis: Mitchell and Vavasis [52] and Ruppert [64] use functions describing the typical local size to bound the number of elements in the meshes their algo... |

23 | Node Nested Multigrid with Delaunay Coarsening
- Guillard
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(Show Context)
Citation Context ... independence: if u1; u2 2 U then (u1; u2) 62 E, and 2. maximality: no node v 2 V can be added to U without violating independence, i.e. all nodes v 2 V n U are incident to some node u 2 U . Guillard =-=[34]-=- suggests a combination of Delaunay based triangulation with an MIS-based coarsening scheme. Nodes on the input PSLG (the boundary) segments are at higher priority, and are considered first for inclus... |

21 | An Advancing Front Delaunay Triangulation Algorithm Designed for Robustness
- Mavriplis
- 1995
(Show Context)
Citation Context ...xplicit functions is using background grids, with function values at the grid nodes interpolated to interior nodes. The background grid can be unstructured [58, 35], Cartesian [59], or quadtree based =-=[45]-=-. ffl In mesh size analysis: Mitchell and Vavasis [52] and Ruppert [64] use functions describing the typical local size to bound the number of elements in the meshes their algorithms produce. Mitchell... |

20 | Multigrid Techniques for Unstructured Meshes
- Mavriplis
- 1995
(Show Context)
Citation Context ...to form a new larger cell. As in the MIS-based schemes, the order by which the volumes are traversed can be prioritized to proceed from the boundary cells inward, or in an advanced front fashion. See =-=[72, 48, 46, 40]-=-. 74s5.2. PREVIOUS WORK Figure 5.3: Repeated applications of MIS can degrade the aspect ratio. 5.2.2 Repeated degradation: problems of topological coarsening MIS on the 1D skeleton is very successful ... |