On a Refinement-Free Calderón Multiplicative Preconditioner for the Electric Field Integral Equation
This work addresses the need for efficient and stable preconditioning of the EFIE in computational electromagnetics, offering a simpler implementation than existing Calderón preconditioners.
The paper presents a Calderón preconditioner for the electric field integral equation that avoids barycentric mesh refinement and Buffa-Christiansen functions, yielding a Hermitian positive definite system solvable with CG. It is immune to low-frequency and dense-discretization breakdown and remains stable on multiply connected geometries, with numerical results showing effectiveness on canonical and multi-scale problems.
We present a Calderón preconditioner for the electric field integral equation (EFIE), which does not require a barycentric refinement of the mesh and which yields a Hermitian, positive definite (HPD) system matrix allowing for the usage of the conjugate gradient (CG) solver. The resulting discrete equation system is immune to the low-frequency and the dense-discretization breakdown and, in contrast to existing Calderón preconditioners, no second discretization of the EFIE operator with Buffa-Christiansen (BC) functions is necessary. This preconditioner is obtained by leveraging on spectral equivalences between (scalar) integral operators, namely the single layer and the hypersingular operator known from electrostatics, on the one hand, and the Laplace-Beltrami operator on the other hand. Since our approach incorporates Helmholtz projectors, there is no search for global loops necessary and thus our method remains stable on multiply connected geometries. The numerical results demonstrate the effectiveness of this approach for both canonical and realistic (multi-scale) problems.