Hierarchical bases preconditioners for a conformingly discretized combined field integral equation operator

It is shown how hierarchical bases preconditioners constructed for stabilizing the electric field integral equation (EFIE) operator can be used for stabilizing the combined field integral equation (CFIE) operator. Since the CFIE operator consists of the EFIE and magnetic field integral equation (MFIE) operator, also the CFIE operator is affected by the low-frequency and the dense-discretization breakdown of the EFIE operator. We show that the hierarchical basis preconditioner for the EFIE operator must not directly be applied to the CFIE operator, since this would introduce a new ill-conditioning. Instead, Helmholtz projectors are used in a left-preconditioning scheme for implicitly regularizing the solenoidal Helmholtz subspace of the CFIE operator, while the hierarchical stars are used for the regularization of the non-solenoidal Helmholtz subspace. Numerical results corroborate the presented theory.