The rise in the use of high-frequency electromagnetic waves has highlighted growing needs and difficulties in the numerical simulation of electromagnetism. Many numerical solvers have been developed to face such problems, as the Finite Difference and Finite Element methods. Yet, considering the large domain dimensions of current time-harmonic frameworks (in terms of wavelengths), such methods lead to linear systems whose direct resolution is memory-prohibitive, and which are poorly adapted to iterative resolution. On the contrary, Trefftz methods, which can be interpreted as a Discontinuous Galerkin formulation whose basis functions are actual solutions of the considered PDE, are able to deal with such calculation scenes [Pernet 2023]. However, the classic choice of plane waves is used to implying ill-conditioned systems, leading us to consider quasi-Trefftz methods [Barucq 2017]: they rely on computed approximate Maxwell solutions, taken as new basis functions. The local solver choice is then essential and multiple methods may be considered. In particular, the Flux Reconstruction (FR) method [Huynh 2007] has successfully established itself for hyperbolic problems in the Computational Fluid Dynamics field. In this talk, we propose an analysis of FR in a time-harmonic framework, with a priori explicit error estimates and optimised correction polynomial function considerations for the 1D wave equation. Besides, this work will be transposed to the 3D time-harmonic Maxwell equations, and adapted as a local solver in a quasi-Trefftz approach. The properties and interests of the methods will be illustrated with numerical examples in HPC context.