Numerics for Helmholtz equation with Microlocal Bases – NuHeMiBa

This project will develop innovative numerical methods to solve the Helmholtz equation, which allow a large reduction of the dimension of the discrete problem at high-frequencies, compared to the usual Finite Element Methods. Unlike alternative approaches such as Boundary Element Methods or Trefftz-like methods, the proposed approach will be perfectly valid for variable coefficients.?

The underlying idea, inspired by microlocal analysis, is to take full advantage of the microlocalization properties of the solution in phase space, that is, localization both in position and Fourier space. Indeed, in many practical problems (such as plane-wave scattering), the solution of the Helmholtz equation is microlocalized near a hypersurface. We will take advantage of this property by building discretization spaces spanned by finitely many functions that are microlocalized near this hypersurface.

While I have obtained preliminary results in an idealized setting (considering the Helmholtz equation in the whole space with smooth coefficients), many challenges have to be faced to make this method applicable in real-life problems. Such challenges, which will be faced in this project, include considering boundaries and discontinuous coefficients, proving the convergence of numerical schemes as the Galerkin problem, undestanding the conditioning of the resulting discrete system and its resolution using iterative methods such as GMRES.