### Abstract

We prove some new estimates for the convergence of multigrid algorithms applied to nonsymmetric and indefinite elliptic boundary value problems. We provide results for the so-called 'symmetric' multigrid schemes. We show that for the variable v-cycle and the m;-cycle schemes, multigrid algorithms with any amount of smoothing on the finest grid converge at a rate that is independent of the number of levels or unknowns, provided that the initial grid is sufficiently fine. We show that the v-cycle algorithm also converges (under appropriate assumptions on the oarsest grid) but at a rate which may deteriorate as the number of levels increases. This deterioration for the m-cycle may occur even in the case of full elliptic regularity. Finally, the results of numerical experiments are given which illustrate the convergence behavior suggested by the theory.

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

Pages (from-to) | 389-414 |

Number of pages | 26 |

Journal | Mathematics of Computation |

Volume | 51 |

Issue number | 184 |

DOIs | |

State | Published - Jan 1 1988 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Computational Mathematics
- Applied Mathematics

### Cite this

*Mathematics of Computation*,

*51*(184), 389-414. https://doi.org/10.1090/S0025-5718-1988-0930228-6

}

*Mathematics of Computation*, vol. 51, no. 184, pp. 389-414. https://doi.org/10.1090/S0025-5718-1988-0930228-6

**The analysis of multigrid algorithms for nonsymmetric and indefinite elliptic problems.** / Bramble, James H.; Pasciak, Joseph E.; Xu, Jinchao.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The analysis of multigrid algorithms for nonsymmetric and indefinite elliptic problems

AU - Bramble, James H.

AU - Pasciak, Joseph E.

AU - Xu, Jinchao

PY - 1988/1/1

Y1 - 1988/1/1

N2 - We prove some new estimates for the convergence of multigrid algorithms applied to nonsymmetric and indefinite elliptic boundary value problems. We provide results for the so-called 'symmetric' multigrid schemes. We show that for the variable v-cycle and the m;-cycle schemes, multigrid algorithms with any amount of smoothing on the finest grid converge at a rate that is independent of the number of levels or unknowns, provided that the initial grid is sufficiently fine. We show that the v-cycle algorithm also converges (under appropriate assumptions on the oarsest grid) but at a rate which may deteriorate as the number of levels increases. This deterioration for the m-cycle may occur even in the case of full elliptic regularity. Finally, the results of numerical experiments are given which illustrate the convergence behavior suggested by the theory.

AB - We prove some new estimates for the convergence of multigrid algorithms applied to nonsymmetric and indefinite elliptic boundary value problems. We provide results for the so-called 'symmetric' multigrid schemes. We show that for the variable v-cycle and the m;-cycle schemes, multigrid algorithms with any amount of smoothing on the finest grid converge at a rate that is independent of the number of levels or unknowns, provided that the initial grid is sufficiently fine. We show that the v-cycle algorithm also converges (under appropriate assumptions on the oarsest grid) but at a rate which may deteriorate as the number of levels increases. This deterioration for the m-cycle may occur even in the case of full elliptic regularity. Finally, the results of numerical experiments are given which illustrate the convergence behavior suggested by the theory.

UR - http://www.scopus.com/inward/record.url?scp=84966234078&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84966234078&partnerID=8YFLogxK

U2 - 10.1090/S0025-5718-1988-0930228-6

DO - 10.1090/S0025-5718-1988-0930228-6

M3 - Article

VL - 51

SP - 389

EP - 414

JO - Mathematics of Computation

JF - Mathematics of Computation

SN - 0025-5718

IS - 184

ER -