## Algebraic convergence for anisotropic edge elements in polyhedral domains

*
Annalisa Buffa,
Martin Costabel,
Monique Dauge*

We study approximation errors for the
**h-version**
of Nédélec edge elements on anisotropically
refined meshes in polyhedra. Both tetrahedral and hexahedral elements are considered, and the
emphasis is on obtaining optimal convergence rates in the
**H(curl)** norm for higher order elements.

Two types of estimates are presented:

1. ** Interpolation ** error estimates for functions in
anisotropic weighted Sobolev spaces. Here we consider not only the
**H(curl)**-conforming
Nédélec elements, but also the
**H(div)**-conforming Raviart-Thomas elements which appear
naturally in the discrete version of the de Rham complex. Our technique is to transport error
estimates from the reference element to the physical element via highly anisotropic coordinate
transformations.

2. ** Galerkin ** error estimates for the standard
**H(curl)** approximation
of time harmonic Maxwell equations. Here we use the anisotropic weighted Sobolev regularity of the
solution on domains with three-dimensional edges and corners, see the
**Note CRAS**.

We also prove the discrete compactness
property needed for the convergence of the Maxwell eigenvalue problem. Our results generalize those
of
[
Nicaise, *SIAM J. Numer. Anal.* 2001]
to the case of polyhedral corners and higher order elements.

November 2003.

Technical report, *Instituto di Matematica Applicata e Tecnologie Informatiche
CNR,* N. ** 28-PV ** 2003.

*Numerische Mathematik,* **101 (2005)** 29-65.