On a refinement-free Calderón multiplicative preconditioner for the electric field integral equation

• Published in: Journal of computational physics Vol. 376; pp. 1232 - 1252
• Main Authors: Adrian, S.B., Andriulli, F.P., Eibert, T.F.
• Format: Journal Article
• Language: English
• Published: Cambridge Elsevier Inc 01-01-2019; Elsevier Science Ltd; Elsevier
• Subjects: Calderón preconditioning; Computational Physics; Discrete element method; Discretization; Electric field integral equation (EFIE); Electric fields; Electrostatics; Helmholtz equations; Hierarchical basis; Integral equations; Mathematical analysis; Matrix methods; Multilevel; Operators (mathematics); Physics; Preconditioner; Projectors; Wavelet; Wavelet; Electric field integral equation (EFIE); Preconditioner; Hierarchical basis; Multilevel; Calderón preconditioning; wavelet; hierarchical basis; preconditioner; multilevel
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2018.10.009

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.
•A well-conditioned electric field integral equation (EFIE).•EFIE system is Hermitian and positive definite.•No second discretization of the EFIE with dual basis functions.•No need for cycle detection on multiply connected geometries.