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Nodal Auxiliary Space Preconditioning in H(curl) and H(div) spaces

by R. Hiptmair and J. Xu

(Report number 2006-09)

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)- and H(div) elliptic variational problems. The preconditioners exclusively rely on solvers for discrete Poisson problems. £We prove mesh independent effectivity of the preconditioners by appealing to the abstract theory of auxiliary space preconditioning. The main tool are discrete analogues of so called regular decomposition results in the function spaces H(curl)- and H(div). Our preconditioner for H(curl) space is similar to an algorithm proposed in (R. Beck, Algebraic multigrid by component splitting for edge elements on simplicial triangulations, Techn. Report SC 99-40, ZIB, Berling, Germany, 1999)

Keywords:

@Techreport{HX06_358,
  author = {R. Hiptmair and J. Xu},
  title = {Nodal Auxiliary Space Preconditioning in H(curl) and H(div) spaces},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2006-09},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2006/2006-09.pdf },
  year = {2006}
}

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