### Abstract

This paper presents a strength of connection measure for algebraic multilevel algorithms for a class of linear systems corresponding to the graph Laplacian on a general graph. The coarsening in the multilevel algorithm is based on partitioning in subgraphs (using matching) of the underlying graph. Our idea is to define a local measure of the quality of the matching that follows from a commutative diagram we introduce, whose maximum gives an upper bound on the stability (energy seminorm) of the orthogonal projection on the coarse space. As an application, we focus on utilizing this measure as a tool for constructing coarse spaces for anisotropic diffusion problems. Specifically, we consider the diffusion equation with grid aligned as well as non-grid-aligned anisotropies in the diffusion coefficient and show that the strength of connection measure is able to appropriately capture the correct anisotropic behavior in both cases. We then study a coarsening algorithm that uses this measure in a greedy strategy to find the subgraph partitioning (set of aggregates). The process forms an initial set of subgraphs, each consisting of a single vertex, and then adds vertices to these subgraphs corresponding to the local direction of the anisotropy as determined by the proposed measure.

Original language | English (US) |
---|---|

Pages (from-to) | 279-295 |

Number of pages | 17 |

Journal | Numerical Linear Algebra with Applications |

Volume | 19 |

Issue number | 2 |

DOIs | |

State | Published - Mar 1 2012 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Applied Mathematics

### Cite this

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*Numerical Linear Algebra with Applications*, vol. 19, no. 2, pp. 279-295. https://doi.org/10.1002/nla.1804

**An algebraic multilevel method for anisotropic elliptic equations based on subgraph matching.** / Brannick, James Joseph; Chen, Yao; Zikatanov, Ludmil Tomov.

Research output: Contribution to journal › Article

TY - JOUR

T1 - An algebraic multilevel method for anisotropic elliptic equations based on subgraph matching

AU - Brannick, James Joseph

AU - Chen, Yao

AU - Zikatanov, Ludmil Tomov

PY - 2012/3/1

Y1 - 2012/3/1

N2 - This paper presents a strength of connection measure for algebraic multilevel algorithms for a class of linear systems corresponding to the graph Laplacian on a general graph. The coarsening in the multilevel algorithm is based on partitioning in subgraphs (using matching) of the underlying graph. Our idea is to define a local measure of the quality of the matching that follows from a commutative diagram we introduce, whose maximum gives an upper bound on the stability (energy seminorm) of the orthogonal projection on the coarse space. As an application, we focus on utilizing this measure as a tool for constructing coarse spaces for anisotropic diffusion problems. Specifically, we consider the diffusion equation with grid aligned as well as non-grid-aligned anisotropies in the diffusion coefficient and show that the strength of connection measure is able to appropriately capture the correct anisotropic behavior in both cases. We then study a coarsening algorithm that uses this measure in a greedy strategy to find the subgraph partitioning (set of aggregates). The process forms an initial set of subgraphs, each consisting of a single vertex, and then adds vertices to these subgraphs corresponding to the local direction of the anisotropy as determined by the proposed measure.

AB - This paper presents a strength of connection measure for algebraic multilevel algorithms for a class of linear systems corresponding to the graph Laplacian on a general graph. The coarsening in the multilevel algorithm is based on partitioning in subgraphs (using matching) of the underlying graph. Our idea is to define a local measure of the quality of the matching that follows from a commutative diagram we introduce, whose maximum gives an upper bound on the stability (energy seminorm) of the orthogonal projection on the coarse space. As an application, we focus on utilizing this measure as a tool for constructing coarse spaces for anisotropic diffusion problems. Specifically, we consider the diffusion equation with grid aligned as well as non-grid-aligned anisotropies in the diffusion coefficient and show that the strength of connection measure is able to appropriately capture the correct anisotropic behavior in both cases. We then study a coarsening algorithm that uses this measure in a greedy strategy to find the subgraph partitioning (set of aggregates). The process forms an initial set of subgraphs, each consisting of a single vertex, and then adds vertices to these subgraphs corresponding to the local direction of the anisotropy as determined by the proposed measure.

UR - http://www.scopus.com/inward/record.url?scp=84862795443&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84862795443&partnerID=8YFLogxK

U2 - 10.1002/nla.1804

DO - 10.1002/nla.1804

M3 - Article

AN - SCOPUS:84862795443

VL - 19

SP - 279

EP - 295

JO - Numerical Linear Algebra with Applications

JF - Numerical Linear Algebra with Applications

SN - 1070-5325

IS - 2

ER -