Variational theory and domain decomposition for nonlocal problems

Burak Aksoylu, Michael L. Parks

Research output: Contribution to journalArticlepeer-review

87 Scopus citations

Abstract

In this article we present the first results on domain decomposition methods for nonlocal operators. We present a nonlocal variational formulation for these operators and establish the well-posedness of associated boundary value problems, proving a nonlocal Poincaré inequality. To determine the conditioning of the discretized operator, we prove a spectral equivalence which leads to a mesh size independent upper bound for the condition number of the stiffness matrix. We then introduce a nonlocal two-domain variational formulation utilizing nonlocal transmission conditions, and prove equivalence with the single-domain formulation. A nonlocal Schur complement is introduced. We establish condition number bounds for the nonlocal stiffness and Schur complement matrices. Supporting numerical experiments demonstrating the conditioning of the nonlocal one- and two-domain problems are presented.

Original languageEnglish
Pages (from-to)6498-6515
Number of pages18
JournalApplied Mathematics and Computation
Volume217
Issue number14
DOIs
StatePublished - Mar 15 2011
Externally publishedYes

Keywords

  • Condition number
  • Domain decomposition
  • Nonlocal operators
  • Nonlocal Poincaré inequality
  • Nonlocal Schur complement
  • Nonlocal substructuring
  • p-Laplacian
  • Peridynamics

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