### Abstract

Multilevel Schwarz methods are developed for a conforming finite element approximation of second order elliptic problem. We focus on problems in three dimensions with possibly large jumps in the coefficients across the interface separating the subregions. We establish a condition number estimate for the iterative operator, which is independent of the coefficients, and grows at most as the square of the number of levels. We also characterize a class of distributions of the coefficients, called quasi-monotone, for which the weighted L^{2}-projection is stable and for which we can use the standard piecewise linear functions as a coarse space. In this case, we obtain optimal methods, i.e. bounds which are independent of the number of levels and subregions. We also design and analyze multilevel methods with new coarse spaces given by simple explicit formulas. We consider nonuniform meshes and conclude by an analysis of multilevel iterative substructuring methods.

Original language | English (US) |
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Pages (from-to) | 313-348 |

Number of pages | 36 |

Journal | Numerische Mathematik |

Volume | 72 |

Issue number | 3 |

DOIs | |

State | Published - Jan 1996 |

### ASJC Scopus subject areas

- Computational Mathematics
- Applied Mathematics

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## Cite this

*Numerische Mathematik*,

*72*(3), 313-348. https://doi.org/10.1007/s002110050172