Title:
A wavelet approximation method for the integral equations of an antenna
problem
Abstract:
The radiation of electromagnetic waves by a printed antenna can be
described by the electrical field integral equation on an open surface
in $\R^3$. A recently developed reformulation [1,2] of this system of
boundary integral equations using Hodge decompositions on the surface
leads to a system that allows to show stability of Galerkin
approximations using standard $C^0$ nodal finite elements.
Simple piecewise linear elements on uniformly refined triangular meshes
permit the construction of biorthogonal wavelet bases that lead to
matrix compression and preconditioning methods giving rise to a solution
algorithm of almost linear complexity. Both for the Hodge decompositions
and the compression estimates, particular care has to be taken of the
boundary of the open surface.
In the talk, theoretical results from [2] about this method will be
presented. Recent computations for a small array of antennas at the
electronics laboratory of the INSA (Rennes) emplyed similar fast
algorithms, allowing for the first time the use of standard workstations
for such problems.
[1] A. Buffa, M. Costabel, C. Schwab: Boundary element methods for
Maxwell's equations on non-smooth domains. Research Report No. 2001-01,
SAM ETHZ, Z\"urich 2001. To appear in Numer. Math.
[2] C. Safa: R\'esolution rapide d'\'equations int\'egrales pour un
probl\`eme d'antennes par des m\'ethodes d'ondelettes. Thesis IRMAR
Univ. Rennes 1, 2001, in preparation.