The Helmholtz equation modelizes the propagation of a time-harmonic wave.
It has received much attention since it is widely employed in applications,
but still challenging to numerically simulate in the high-frequency regime.

In this seminar, we focus on acoustic waves for the sake of simplicity
and consider finite element discretizations. The main goal of the
presentation is to highlight the improved performance of high order
methods (as compared to linear finite elements) when the frequency is large.

We first treat the zero-frequency case, that corresponds to the well-studied
Poisson equation. We take advantage of this classical setting to present
central concepts of the finite element theory such as quasi-optimality
and interpolation error.

The second part of the seminar is devoted to the high-frequency case.
We show that without restrictive assumptions on the mesh size,
the finite element method is unstable, and quasi-optimality is lost.
We provide a detailed analysis, as well as numerical examples, which
show that higher order methods are less affected by this phenomena,
and thus more suited to discretize high-frequency problems.

Before drawing our main conclusions, we briefly discuss advanced topics,
such as the use of unfitted meshes in highly heterogeneous media
and mesh refinements around re-entrant corners.