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Stable multilevel splittings of boundary edge element spaces

by R. Hiptmair and S.-P. Mao

(Report number 2011-28)

Abstract
We establish the stability of nodal multilevel decompositions of lowest-order conforming boundary element subspaces of the trace space H^(-/frac(1) (2)) (div T,T) of H(curl, Omega) on boundaries of triangulated Lipschitz polyhedra. The decompositions are based on nested triangular meshes created by regular refinement and the stability bounds are uniform in the number of refinement levels. The main tool is the general theory of [P. Oswald, Interface preconditioners and multilevel extension operators, in Proc. 11th Intern. Conf. on Domain Decomposition Methods, London 1998, pp. 96–103] that teaches, when stability of decompositions of boundary element spaces with respect to trace norms can be inferred from corresponding stability results for finite element spaces. H(curl, Omega)-stable discrete extension operators are instrumental in this. Stable multilevel decompositions immediately spawn subspace correction preconditioners whose performance will not degrade on very fine surface meshes. Thus, the results of this article demonstrate how to construct optimal iterative solvers for the linear systems of equations arising from the Galerkin edge element discretization of boundary integral equations for eddy current problems.

Keywords:

@Techreport{HM11_67,
  author = {R. Hiptmair and S.-P. Mao},
  title = {Stable multilevel splittings of boundary edge element spaces},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2011-28},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2011/2011-28.pdf },
  year = {2011}
}

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