Schwarz methods of neumann‐neumann type for three‐dimensional elliptic finite element problems

Maksymilian Dryja, Olof B. Widlund

Research output: Contribution to journalArticle

Abstract

Several domain decomposition methods of Neumann‐Neumann type are considered for solving the large linear systems of algebraic equations that arise from discretizations of elliptic problems by finite elements. We will only consider problems in three dimensions. Several new variants of the basic algorithm are introduced in a Schwarz method framework that provides tools which have already proven very useful in the design and analysis of other domain decomposition and multi‐level methods. The Neumann‐Neumann algorithms have several advantages over other domain decomposition methods. The subregions, which define the subproblems, only share the boundary degrees of freedom with their neighbors. The subregions can also be of quite arbitrary shape and many of the major components of the preconditioner can be constructed from subprograms available in standard finite element program libraries. In its original form, however, the algorithm lacks a mechanism for global transportation of information and its performance therefore suffers when the number of subregions increases. In the new variants of the algorithms, considered in this paper, the preconditioners include global components, of low rank, to overcome this difficulty. Bounds are established for the condition number of the iteration operator, which are independent of the number of subregions, and depend only polylogarithmically on the number of degrees of freedom of individual local subproblems. Results are also given for problems with arbitrarily large jumps in the coefficients across the interfaces separating the subregions. ©1995 John Wiley & Sons, Inc.

Original languageEnglish (US)
Pages (from-to)121-155
Number of pages35
JournalCommunications on Pure and Applied Mathematics
Volume48
Issue number2
DOIs
StatePublished - 1995

Fingerprint

Schwarz Methods
Domain Decomposition Method
Finite Element
Domain decomposition methods
Three-dimensional
Preconditioner
Degree of freedom
Multilevel Methods
Condition number
Algebraic Equation
Elliptic Problems
Linear systems
Three-dimension
Jump
Discretization
Linear Systems
Decomposition
Iteration
Arbitrary
Coefficient

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Schwarz methods of neumann‐neumann type for three‐dimensional elliptic finite element problems. / Dryja, Maksymilian; Widlund, Olof B.

In: Communications on Pure and Applied Mathematics, Vol. 48, No. 2, 1995, p. 121-155.

Research output: Contribution to journalArticle

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