On the Hierarchical Preconditioning of the Combined Field Integral Equation
This work addresses the need for efficient preconditioning of the Combined Field Integral Equation for electromagnetic scattering problems, particularly on unstructured meshes where hierarchical solenoidal bases are unavailable.
The paper extends hierarchical preconditioning from the Electric Field Integral Equation to the Combined Field Integral Equation, proposing a new scheme that uses a Helmholtz projector for the solenoidal subspace and a hierarchical non-solenoidal basis for the non-solenoidal subspace, resulting in a well-conditioned system as confirmed by numerical results.
This paper analyzes how hierarchical bases preconditioners constructed for the Electric Field Integral Equation (EFIE) can be effectively applied to the Combined Field Integral Equation (CFIE). For the case where no hierarchical solenoidal basis is available (e.g., on unstructured meshes), a new scheme is proposed: the CFIE is implicitly preconditioned on the solenoidal Helmholtz subspace by using a Helmholtz projector, while a hierarchical non-solenoidal basis is used for the non-solenoidal Helmholtz subspace. This results in a well-conditioned system. Numerical results corroborate the presented theory.