Nodal auxiliary space preconditioning in H(curl) and H(div) spaces

Ralf Hiptmair, Jinchao Xu

Research output: Contribution to journalArticle

157 Scopus citations

Abstract

In this paper, we develop and analyze a general approach to preconditioning linear systems of equations arising from conforming finite element discretizations of H(curl, Ω)- and H(div, Ω)-elliptic variational problems. The preconditioners exclusively rely on solvers for discrete Poisson problems. We prove mesh-independent effectivity of the preconditioners by using the abstract theory of auxiliary space preconditioning. The main tools are discrete analogues of so-called regular decomposition results in the function spaces H(curl, Ω) and H(div, Ω). Our preconditioner for H(curl, Ω) is similar to an algorithm proposed in [R. Beck, Algebraic Multigrid by Component Splitting for Edge Elements on Simplicial Triangulations, Tech. rep. SC 99-40, ZIB, Berlin, Germany, 1999].

Original languageEnglish (US)
Pages (from-to)2483-2509
Number of pages27
JournalSIAM Journal on Numerical Analysis
Volume45
Issue number6
DOIs
StatePublished - Dec 1 2007

All Science Journal Classification (ASJC) codes

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

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