Numerical approximation of time-harmonic and time-dependent wave propagation problems

The present talk deals with numerical techniques to discretize wave propagation problems. In the first part of the talk I will focus on the interior Helmholtz equation as well as on scattering problems. Due to oscillations in the analytical solutions, accurate finite element approximations are computationally expensive and time-consuming, already for moderate frequencies. Therefore, when responses at many frequencies are of interest, their direct computation is unaffordable. I will discuss a model order reduction method aimed at reducing the computational costs by producing an approximation of (some functional of) the frequency response map.
In the second part of the talk, I will present a space-time discretization for the scalar-valued dissipative wave equation in two- and three-dimensions. It is a structured approach based on the Virtual Element Method for space discretization coupled with the Discontinuous Galerkin finite element method for time integration. I will discuss the theoretical analysis of the scheme and present some numerical examples to demonstrate its practical capabilities. Finally, I will conclude with some perspectives and future developments.