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

Maksymilian Dryja, Marcus V. Sarkis, Olof B. Widlund

Research output: Contribution to journalArticle

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 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.

Original languageEnglish (US)
Pages (from-to)313-348
Number of pages36
JournalNumerische Mathematik
Volume72
Issue number3
StatePublished - Jan 1996

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Schwarz Methods
Multilevel Methods
Discontinuous Coefficients
Iterative methods
Elliptic Problems
Three-dimension
Coefficient
Iterative Substructuring
Non-uniform Mesh
Second-order Elliptic Problems
Piecewise Linear Function
Condition number
Finite Element Approximation
Explicit Formula
Monotone
Jump
Projection
Operator
Estimate

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics
  • Computational Mathematics

Cite this

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

In: Numerische Mathematik, Vol. 72, No. 3, 01.1996, p. 313-348.

Research output: Contribution to journalArticle

Dryja, Maksymilian ; Sarkis, Marcus V. ; Widlund, Olof B. / Multilevel Schwarz methods for elliptic problems with discontinuous coefficients in three dimensions. In: Numerische Mathematik. 1996 ; Vol. 72, No. 3. pp. 313-348.
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