A Preconditioner for the Electric Field Integral Equation Based on Calderon Formulas

We describe a preconditioning technique for the Galerkin approximation of the electric field integral equation (EFIE), which arises in the scattering theory for harmonic electromagnetic waves. It is based on a discretization of the Calderon formulas and the Helmholtz decomposition. We prove several properties of the method, in particular that it produces a variational solution on a subspace of the Galerkin space for which we have an LBB inf-sup condition. When the Krylov spaces associated with the continuous operators are nondegenerate we prove that the discrete Krylov spaces converge as the mesh refinement goes to zero; when, moreover, the EFIE is nondegenerate on the continuous Krylov spaces, the discrete Krylov iterates converge towards the continuous ones. We also argue that one might expect the continuous Krylov iterates to exhibit superlinear convergence of the form $n \mapsto C^n(n!)^{-\alpha}$ for some C > 0 and $\alpha>0$. Finally, we illustrate the theory with numerical experiments.