Exploring Algebraic Preconditioning of EFIE Matrices Arising from Higher Order Additive Singular Bases

Currently there are no operator-dependent preconditioners (for example of the Calderón type) to handle matrices obtained using high-order singular vector bases. So this letter has a dual purpose. The first is to show that the electric field integral equation (EFIE) discretized by high-order singular bases can be quickly solved iteratively using special general-purpose algebraic preconditioners. The second is to demonstrate that the results obtained with the fast solver have the same accuracy as those obtained using classical direct solution methods. The algebraic preconditioner specifically considered here has been used elsewhere to efficiently solve problems with several million unknowns. Thus, in light of the dual purpose and without loss of generality, we use as benchmarks medium-sized test problems involving singular induced currents because, for these problems, the preconditioned solutions can be compared with those obtained by direct methods, which are notoriously unsuitable for solving very large, ill-conditioned problems. In particular, to demonstrate that our approach correctly models the singular behavior of fields in the near-field region, we report several numerical results for current components induced by plane waves on infinitely thin flat plates. On the edges of these plates the current component parallel to the edges can be unlimited (i.e. going to infinity), while the component normal to the edges must vanish. This behavior is correctly modeled by our singular bases when necessary and is not corrupted by the fast solver, which demonstrates the effectiveness and robustness of the singular bases and the preconditioner used.