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Auxiliary Space Preconditioners for SIP-DG Discretizations of H(curl)-elliptic Problems with Discontinuous Coefficients

by B. Ayuso de Dios and R. Hiptmair and C. Pagliantini

(Report number 2015-14)

Abstract
We propose a family of preconditioners for linear systems of equations arising from a piecewise polynomial symmetric Interior Penalty Discontinuous Galerkin (IP-DG) discretization of H(curl,$\Omega$)-elliptic boundary value problems on conforming meshes. The design and analysis of the proposed preconditioners relies on the auxiliary space method (ASM) employing an auxiliary space of H(curl,$\Omega$)-conforming finite element functions together with a relaxation technique (local smoothing). On simplicial meshes, the proposed preconditioner enjoys asymptotic optimality with respect to mesh refinement. It is also robust with respect to jumps in the coefficients $\nu$ and $\beta$ in the second- and zeroth-order parts of the operator, respectively, except when the problem changes from curl-dominated to reaction-dominated and vice versa. On quadrilateral/hexahedral meshes some of the proposed ASM solvers may fail, since the related H(curl,$\Omega$)-conforming finite element space does not provide a spectrally accurate discretization. Extensive numerical experiments are included to verify the theory and assess the performance of the preconditioners.

Keywords: Auxiliary Space Preconditioning, Discontinuous Galerkin, Symmetric Interior Penalty, H(curl), Discontinuous Coefficients

@Techreport{AHP15_604,
  author = {B. Ayuso de Dios and R. Hiptmair and C. Pagliantini},
  title = {Auxiliary Space Preconditioners for SIP-DG Discretizations of H(curl)-elliptic Problems with Discontinuous Coefficients},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2015-14},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2015/2015-14.pdf },
  year = {2015}
}

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