Integral equation solution of low frequency scattering with a conditioning that grows only Logarithmically with the number of unknowns

A detailed analysis of the electric field integral equation (EFIE) at low frequencies is presented which is based on the Sobolev space mapping properties of the EFIE. The theory shows that in the low-frequency regime an EFIE with a logarithmic condition number growth can be obtained. This, provided that both the solenoidal and non-solenoidal parts of the EFIE are properly regularized. The regularization is obtained by introducing a new set of loop hierarchical basis functions and the condition number growth for the proposed scheme is provably O(log² N). Numerical tests confirm the results obtained by the theory.