A hermitian and well-conditioned EFIE for fast iterative and direct solvers

We present a well-conditioned discretization of the electric field integral equation (EFIE) which is Hermitian and positive definite at any frequency. Differently from other preconditioning techniques, like hierarchical methods or Calderón schemes, no refinement of the mesh nor dual basis functions are necessary by the new formulation. This results in substantial computational savings since only the standard system matrix on the regular mesh needs to be stored. Moreover, the coexistence of hermitianity, positive definiteness, and well-conditioning properties allow for a fast direct inversion of the matrix in linear complexity. The new formulation is based on the regularization of the EFIE with a conjugated EFIE. The two matrices are linked with suitably chosen differential operators and Gram matrices that prevent the squaring of the condition number, and instead ensure a uniformly well-conditioned system matrix. Numerical results corroborate the theory and show the practical applicability of the new scheme.