Paper detail

On the numerical stability of the least-squares method for the planar scattering by obstacles

The scattering of waves by obstacles in a 2D setting is considered, in particular the computation of the scattered field via the collocation or the least-squares methods. In the case of multiple scattering by smooth obstacles, we prove that the scattered field can be uniformly approximated by sums of multipoles. For a unique obstacle, the choice of the number of points and their positions for the estimation of the error on the border of the scatterer is studied, showing the benefit of using a non-uniform distribution of points dependent on the scatterer and the approximation scheme. In general, using a denser discretization near the singularities of the scattered field does not improve the stability of the method. The analysis can also be used to estimate the discretization size needed to ensure stability given a density of points and an approximation scheme, e.g. in the case of multiple scatterers.