Generalized impedance boundary conditions for thin dielectric coatings with variable thickness

We derive the so-called Generalized Impedance Boundary Conditions (GIBC) that model thin dielectric coatings with variable width located on a perfect electric or magnetic conducting surface. We treat the 2-D electromagnetic problem for both TM and TE polarizations. The expressions of the GIBCs are obtained up to the third order (with respect to the coating width). The order of convergence is numerically validated through various numerical examples using a finite element type method. Particular attention is given to the cases where the inner boundary has corner singularities.