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The hp Version of the Weighted Regularization Method for Maxwell Equations

*Martin Costabel and Monique Dauge*
Slides of the talk given by M.C. at the LMS Durham Symposium:

Computational methods for wave propagation in direct scattering.

15-25 July 2002, Durham, UK

PDF (760 k)

The weighted regularization method allows the approximation of solutions of
time-harmonic Maxwell equations by standard nodal finite elements in
situations where the standard regularization method fails due to strong
non-H1 singularities at reentrant corners and edges.

Since this method is based on a simple variational formulation with a
symmetric positive bilinear form and the energy space is compactly embedded
into L2, one has all the usual tools like Cea's lemma and convergence of
eigenvalues and eigenspaces.

The one remaining theoretical problem concerns the approximation properties
of standard finite elements in the energy space which is defined by the L2
norm for the field and its curl, and a weighted L2 norm for the
divergence. We have convergence proofs for the h version and now also for
the hp version with exponential convergence in the latter case. In the
proofs, we need to use C1 interpolants, a technique that leads to some
restrictions on the allowed meshes.

The slides illustrate the effect of the weight in the regularization term by
results of computations in 2D and 3D domains with reentrant corners. One
observes the predicted exponential convergence of the eigenvalues. In 2D, we
present results on straight and curved L-shaped domains, and in 3D, we show
that the method works well on the "thick L" and on Fichera's corner.

For theoretical background, corner and edge singularities and all that, see
Monique Dauge's
Durham talk .