Low-Frequency Stabilization for the B-Spline-Based Isogeometric Discretization of the Electric Field Integral Equation

In order to low-frequency stabilize the electric field integral equation (EFIE) when discretized with divergence conforming B-spline-based basis and testing functions in an isogeometric approach, we propose a corresponding quasi-Helmholtz preconditioner. To this end, we derive i) a loop-star decomposition for the B-spline basis in the form of sparse mapping matrices applicable to arbitrary polynomial orders of the basis as well as to open and closed geometries described by single-patch or multipatch parametric surfaces (as an example, nonuniform rational B-splines (NURBS) surfaces are considered). Based on the loop-star analysis, we show ii) that quasi-Helmholtz projectors can be defined efficiently. This renders the proposed low-frequency stabilization directly applicable to multiply-connected geometries without the need to search for global loops and results in better-conditioned system matrices compared with directly using the loop-star basis. Numerical results demonstrate the effectiveness of the proposed approach.