In this work, we extend the standard quasi-Helmholtz filters to high-order discretizations in the context of the boundary element method. This generalization allows preconditioning integral equations such as the electric field integral equation (EFIE) in the h-refinement regime while preserving the accuracy gain of high-order discretizations. The theoretical framework will be corroborated by numerical results that validate the effectiveness of the proposed strategy when stabilizing the refinement-dependent spectral behavior of the high-order discretized EFIE.

The first of Calderón identities indicates that the operator of the electric field integral equation (EFIE) can potentially act as a preconditioner for itself. However, the discretization of this preconditioning operator via the boundary element method (BEM) requires constructing a dual space commonly defined on a barycentric refinement of the orginal mesh. This contribution introduces a novel Calderón strategy that leverages high-order quasi-Helmholtz projectors. Unlike the standard strategies, which are based on barycentric refinements, a dual space is obtained by combinating functions of higher order. This strategy allows the development of a discretization of the Calderón identity that is suitable for arbitrary order. The well-conditioning of the proposed formulation is investigated, and numerical results are presented to corroborate the theory.

The electric field integral equation (EFIE) is known to suffer from ill-conditioning and numerical instabilities at low frequencies (low-frequency breakdown). A common approach to solve this problem is to rely on the loop and star decomposition of the unknowns. Unfortunatelly, building the loops is challenging in many applications, especially in the presence of junctions. In this work, we investigate the effectiveness of quasi-Helmholtz projector approaches in problems containing junctions for curing the low-frequency breakdown without detecting the global loops. Our study suggests that the performance of the algorithms required to obtain the projectors in the presence of junctions is maintained while keeping constant the number of sheets per junction. Finally, with a sequence of numerical tests, this work shows the practical impact of the technique and its applicability to real case scenarios.