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 .