Microlocal analysis and numerical methods for wave propagation – MicroWave

The aim of the « Microwave » ANR Blanc proposal is to build and implement new simulation techniques using microlocal analysis for pseudo- and paradifferential operators. Until now, microlocal methods were mainly applied to the analysis of Partial Differential Equations, leading to fine mathematical properties. However, their applications to scientific computing have been much less investigated. For this reason, we propose to develop this original and promising direction of research choosing applications related to wave phenomenas. In a first step, our developments will follow two directions. The first direction that will be pursued consists in improving some numerical methods for solving high frequency acoustic and electromagnetic scattering problems governed by the Helmholtz/Maxwell's equations. We propose to focus on volumetric formulations based on variational approaches as well as integral equations formulations. In particular, we will consider the problem of computing an accurate solution to the scattering problem (pollution phenomena) and the acceleration and convergence of iterative solvers (construction of suitable preconditioners). These two difficult problems are known to be strong limitations in simulations codes, and in particular for industrial applications. The members of the MicroWave ANR Project have an excellent knowledge of the subject and have many contributions to the topic. These qualities constitute therefore a solid basis for considering such problems. The second direction concerns the construction of approximations of the Dirichlet-to-Neumann map for some linear and nonlinear Schrödinger equations, time-dependent or stationary, deterministic or stochastic, coupled systems' These approximations, based on microlocal analysis for pseudo- and para-differential operators, will next be applied to the theory of artificial boundary conditions and to domain decomposition methods. In particular, we will analyze the construction of stable discretization schemes for the corresponding problems in the bounded domains. The members of the MicroWave ANR Project have numerous contributions on these topics. This is an important point for the success of our proposal. A third direction, beginning during the third year, consists in developing some applications of the microlocal numerical methods for the Helmholtz/Maxwell's equations to the Schrödinger equations, and conversely. All these approaches will be implemented in the finite element solver code GetDP, developed by Christophe GEUZAINE who is an associated exterior member of the MicroWave ANR Project. Finally, nontrivial examples associated with realistic physical examples will be considered as benchmarks for validating our achievements.