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On the Incompleteness of Integral Equation Formulations for Electro-Magnetic Field Computations

It is shown that integral equation formulations for electromagnetic field problems which amount to compact operators (as found e.g. in boundary integral or spectral domain approaches) are incomplete problem specifications for purposes of numerical solution. They describe problems which are ill-posed in the sense of Hadamard. This difficulty becomes apparent after some implementation specific degree of problem complexity is reached. It is demonstrated that for electromagnetic field eigenvalue problems the consequence is a failure to detect any solution, except eventually spurious solutions if an inadequate discretization procedure is employed. It is concluded that integral equation formulations with compact operators must be accompanied by regularization conditions to yield a complete problem specification. Theoretical results are illustrated by numerical details from full-wave boundary integral equation analysis of a coplanar waveguide with conductors of finite thickness and finite conductivity.