2026/07/22 by T. Chaumont-Frelet, M. Ingremeau, F. Proust
#math.NA #cs.NA #math.AP
We propose a new Galerkin discretization scheme for wave scattering problems that is based on microlocalised basis functions. We show that the proposed method can be made uniformly accurate for large wavenumbers k with a number of degrees of freedom only scaling as kd-1/2, while leading to an essentially sparse linear system. In contrast, finite element methods are known to require a number of degrees of freedom scaling at least as kd to achieve the same property. A similar method based on a Gabor frame was previously introduced by two of the authors, but it was suffering from severe conditioning issues. In the present work, by replacing the Gabor frame by a Wilson basis, we completely alleviate this problem. We rigorously establish error estimates and condition number bounds for the proposed method, and we provide one-dimensional numerical examples illustrating our theoretical findings.