### Abstract

We present a Fourier analysis of multigrid for the two-dimensional discrete convection-diffusion equation. For constant coefficient problems with grid-aligned flow and semi-periodic boundary conditions, we show that the two-grid iteration matrix can be reduced via a set of orthogonal transformations to a matrix containing individual 4×4 blocks. This enables a trivial computation of the norm of the iteration matrix demonstrating rapid convergence in the case of both small and large mesh Peclet numbers, where the streamline-diffusion discretisation is used in the latter case. We also demonstrate that these results are strongly correlated with the properties of the iteration matrix arising from Dirichlet boundary conditions.

Original language | English |
---|---|

Pages (from-to) | 283-306 |

Number of pages | 24 |

Journal | BIT Numerical Mathematics |

Volume | 46 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1 Jun 2006 |

### Fingerprint

### Keywords

- convection-diffusion
- convergence
- multigrid

### Cite this

}

*BIT Numerical Mathematics*, vol. 46, no. 2, pp. 283-306. https://doi.org/10.1007/s10543-006-0052-5

**Fourier analysis of multigrid for a model two-dimensional convection-diffusion equation.** / Elman, Howard C.; Ramage, Alison.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Fourier analysis of multigrid for a model two-dimensional convection-diffusion equation

AU - Elman, Howard C.

AU - Ramage, Alison

PY - 2006/6/1

Y1 - 2006/6/1

N2 - We present a Fourier analysis of multigrid for the two-dimensional discrete convection-diffusion equation. For constant coefficient problems with grid-aligned flow and semi-periodic boundary conditions, we show that the two-grid iteration matrix can be reduced via a set of orthogonal transformations to a matrix containing individual 4×4 blocks. This enables a trivial computation of the norm of the iteration matrix demonstrating rapid convergence in the case of both small and large mesh Peclet numbers, where the streamline-diffusion discretisation is used in the latter case. We also demonstrate that these results are strongly correlated with the properties of the iteration matrix arising from Dirichlet boundary conditions.

AB - We present a Fourier analysis of multigrid for the two-dimensional discrete convection-diffusion equation. For constant coefficient problems with grid-aligned flow and semi-periodic boundary conditions, we show that the two-grid iteration matrix can be reduced via a set of orthogonal transformations to a matrix containing individual 4×4 blocks. This enables a trivial computation of the norm of the iteration matrix demonstrating rapid convergence in the case of both small and large mesh Peclet numbers, where the streamline-diffusion discretisation is used in the latter case. We also demonstrate that these results are strongly correlated with the properties of the iteration matrix arising from Dirichlet boundary conditions.

KW - convection-diffusion

KW - convergence

KW - multigrid

UR - http://www.scopus.com/inward/record.url?scp=33745297120&partnerID=8YFLogxK

U2 - 10.1007/s10543-006-0052-5

DO - 10.1007/s10543-006-0052-5

M3 - Article

AN - SCOPUS:33745297120

VL - 46

SP - 283

EP - 306

JO - BIT Numerical Mathematics

JF - BIT Numerical Mathematics

SN - 0006-3835

IS - 2

ER -