Discrete fractional Sobolev norms for domain decomposition preconditioning

M Arioli, D Kourounis, Daniel Loghin

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Abstract

We present a new approach for preconditioning the interface Schur complement arising in the domain decomposition of second-order scalar elliptic problems. The preconditioners are discrete interpolation norms recently introduced in Arioli & Loghin (2009, Discrete interpolation norms with applications. SIAM J. Numer. Anal., 47, 2924–2951). In particular, we employ discrete representations of norms for the Sobolev space of index 1/2 to approximate the Steklov–Poincaré operators arising from nonoverlapping one-level domain decomposition methods. We use the coercivity and continuity of the Schur complement with respect to the preconditioning norm to derive mesh-independent bounds on the convergence of iterative solvers. We also address the case of nonconstant coefficients by considering the interpolation of weighted spaces and the corresponding discrete norms.
Original languageEnglish
Pages (from-to)318-342
JournalI M A Journal of Numerical Analysis
Volume33
DOIs
Publication statusPublished - 1 Jan 2013

Keywords

  • fractional Sobolev norms;
  • square-root Laplacian
  • interface preconditioners
  • domain decomposition
  • generalized Lanczos algorithm

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