This paper presents a hybrid method called Phase Reduction Finite Element Method as an alternative to the standard finite element method to numerically solve high-frequency time-harmonic scattering problems. We know that the accuracy of the standard FEM is largely reduced by the so-called numerical pollution. The approach presented here is a wise combination of asymptotic approximation with numerical methods: we first approximate the phase of the highly oscillatory solution and then reformulate the initial problem with a new slowly oscillatory envelop which can be accurately approximated by the standard first order finite element method. This procedure has already been studied before, the crucial point remaining in the accuracy of the phase approximation. Here we approximate the phase by solving the eikonal equation with a fast marching method. Numerical simulations have shown improvement in accuracy at very high frequency while using coarser mesh grids. A new low cost discretization rule has been established to bound the numerical pollution for a two-dimensional Dirichlet problem. Experiment has validated the empirical approach, which is a call for further development.

Helmholtz equation, time harmonic, high frequency, finite element, numerical pollution, phase reduction, eikonal equation, fast marching.