Phase reduction models for improving the accuracy of the finite element solution of time-harmonic scattering problems I: General approach and low-order models

This paper introduces a new formulation of high frequency time-harmonic scattering problems in view of a numerical finite element solution. It is well-known that pollution error causes inaccuracies in the finite element solution of short-wave problems. To partially avoid this precision problem, the strategy proposed here consists in firstly numerically computing at a low cost an approximate phase of the exact solution through asymptotic propagative models. Secondly, using this approximate phase, a slowly varying unknown envelope is introduced and is computed using coarser mesh grids. The global procedure is called Phase Reduction. In this first paper, the general theoretical procedure is developed and low-order propagative models are numerically investigated in detail. Improved solutions based on higher order models are discussed showing the potential of the method for further developments.