Preconditioning Strategies and Conformal Discretizations Empowered by High-Order Projectors

We present novel discretization schemes for surface inte- gral equations within the framework of high-order bound- ary element methods. To overcome major limitations in existing high-order techniques, this paper introduces three key contributions: (i) a new conformal testing based on high-order quasi-Helmholtz projectors ensures the adequate high-order convergence of the magnetic field integral equa- tion; (ii) this strategy allows to properly discretize the com- bined field integral equation, resulting for the first time in a resonance-free high-order formulation; and (iii) an origi- nal Calderón-like preconditioner also based on high-order conformal testing stabilizes these formulations for low- frequency and dense-refinement scenarios. This guarantees the rapid convergence of iterative solvers while enabling the efficient use of high-order methods across a wide frequency range and dense-mesh scenarios. Numerical experiments are provided to showcase the effectiveness of our approach