Numerical Analysis of the Exterior Boundary Value Problem for the Time-Harmonic Maxwell Equations by a Boundary Finite Element Method

A general finite element method is applied to compute the skin currents flowing on a perfectly conducting surface when it is illuminated by a time-harmonic incident electromagnetic wave. In this paper, we introduce and study the framework in which the continuous problem can be stated in order to make possible the numerical analysis which will follow in a second part. 0. Introduction. The determination of the diffracted field by a perfectly conducting obstacle T (which is supposed here to be the smooth boundary of a bounded open domain ß') is reduced to that of the surface currents j and charges p on T (cf. e.g. [11], [15], [22]) which satisfy the integral equation (0.1) II/-grad v + iupa\ = -Ue1"0 on T. n is the orthogonal projection on the tangent plane of T; e"10 is the electric part of the incident electromagnetic wave; e and ju are the characteristic constants of the medium in which T is embedded. The time variation is supposed to be e~"°' and is suppressed by linearity; v and a are respectively the scalar and the vector potential of the electric field diffracted by T, respectively created by the surface charges p and the surface currents^': (0.2) v(x)= (G(x,y)p(y)dy(y), ■T a(x)= ¡G(x,y)f(y)dy(y). £ik\x-y\ G(x, y) = -—:-: is the kernel giving the outgoing solutions of the Helmholtz equation; (0.5) k = wv/ëjû is the wave number. Received October 14, 1982. 1980 Mathematics Subject Classification. Primary 65R20; Secondary 78A45. '-1984 American Mathematical Society 0025-5718/84 $1.00 + $.25 per page 29 (0.3)