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We consider a transmission boundary-value problem for the time-harmonic Maxwell equations neglecting displacement currents. The usual transmission conditions, which require the continuity of the tangential components of the electric and magnetic fields across boundaries are slightly modified. For this new problem we show that the uniqueness of the solution depends on the topological properties of the domains under consideration. Finally we obtain existence results by using a boundary integral equation approach.

We consider a transmission boundary-value problem for the time-harmonic Maxwell equations without displacement currents. As transmission conditions we use the continuity of the tangential parts of the magnetic field H and the continuity of the normal components of the magnetization B=müH. This problem, which is posed over all IR3, is then restricted to a bounded domain by introducing artificial boundary conditions. We present uniqueness and existence proofs for this problem using an integral equation approach and compare the results with those obtained in the unbounded case.

We consider two transmission boundary-value problems for the time-harmonic Maxwell equations without displacement currents. For the first problem we use the continuity of the tangential parts of the electric and magnetic fields across material discontinuities as transmission conditions. In the second case the continuity of the tangential components of the electric field E is replaced by the continuity of the normal component of the magnetization B=müH. For this problem existence of solutions is already shown in [6]. If the domains under consideration are not simply connected the solution is not unique. In this paper, we improve the regularity results obtained in [6] and then prove existence and uniqueness theorems for the first problem by extracting its solution out of the set of all solutions of the second problem. Thus we establish a connection between the solutions corresponding to the different transmission boundary conditions.

This paper considers the numerical solution of a transmission boundary-value problem for the time-harmonic Maxwell equations with the help of a special finite volume discretization. Applying this technique to several three-dimensional test problems, we obtain large, sparse, complex linear systems, which are solved by using BiCG, CGS, BiCGSTAB resp., GMRES. We combine these methods with suitably chosen preconditioning matrices and compare the speed of convergence.