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Exponential Convergence of hp-FEM for Maxwell's Equations with Weighted Regularization in Polygonal Domains

by M. Costabel and M. Dauge and Ch. Schwab

(Report number 2004-05)

Abstract
The time-harmonic Maxwell equations do not have an elliptic nature by themselves. Their regularization by a divergence term is a standard tool to obtain equivalent elliptic problems. Nodal finite element discretizations of Maxwell's equations obtained from such a regularization converge to wrong solutions in any non-convex polygon. Modification of the regularization term consisting in the introduction of a weight restores the convergence of nodal FEM, providing optimal convergence rates for the h Version of Finite Elements, [20]. We prove exponential convergence of hp FEM for the weighted regularization of Maxwell's equations in plane polygonal domains provided the hp-FE spaces satisfy a series of axioms. We verify these axioms for several specific families of hp finite element spaces.

Keywords:

@Techreport{CDS04_332,
  author = {M. Costabel and M. Dauge and Ch. Schwab},
  title = {Exponential Convergence of hp-FEM for Maxwell's Equations with Weighted Regularization in Polygonal Domains},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2004-05},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2004/2004-05.pdf },
  year = {2004}
}

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