A failure of a numerical electromagnetics code to match a simple boundary condition

It is shown that it is impossible to satisfy the boundary condition for a perfectly conducting hemisphere-capped dipole without creating a discontinuity in the first derivative of the scattered field on the boundary. Since solutions of Maxwell's equations are analytic in a source-free region, all field derivatives must be continuous in the region where the source-free equations are satisfied. The basis functions used by the generalized multipole technique (GMT) are solutions of Maxwell's source-free equations in a region which includes the scattering surface. Therefore, a GMT solution to the dipole problem can exist only in the sense that the boundary value error approaches zero as the number of basis functions approaches infinity. This in itself is not surprising, but the difficulty of matching the boundary condition at the discontinuity affects the convergence of the technique. For the method-of-moments (MOM) technique, where a source current exists on the scattering surface, it is not clear if a perfect boundary value solution to the dipole problem can be theoretically realized or not.<
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