### 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) |
---|---|

Pages (from-to) | 313-348 |

Number of pages | 36 |

Journal | Numerische Mathematik |

Volume | 72 |

Issue number | 3 |

State | Published - Jan 1996 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics
- Computational Mathematics

### Cite this

*Numerische Mathematik*,

*72*(3), 313-348.

**Multilevel Schwarz methods for elliptic problems with discontinuous coefficients in three dimensions.** / Dryja, Maksymilian; Sarkis, Marcus V.; Widlund, Olof B.

Research output: Contribution to journal › Article

*Numerische Mathematik*, vol. 72, no. 3, pp. 313-348.

}

TY - JOUR

T1 - Multilevel Schwarz methods for elliptic problems with discontinuous coefficients in three dimensions

AU - Dryja, Maksymilian

AU - Sarkis, Marcus V.

AU - Widlund, Olof B.

PY - 1996/1

Y1 - 1996/1

N2 - 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 L2-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.

AB - 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 L2-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.

UR - http://www.scopus.com/inward/record.url?scp=0030556926&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0030556926&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0030556926

VL - 72

SP - 313

EP - 348

JO - Numerische Mathematik

JF - Numerische Mathematik

SN - 0029-599X

IS - 3

ER -