QUASI-LOCAL AND FREQUENCY-ROBUST PRECONDITIONERS FOR THE HELMHOLTZ FIRST-KIND INTEGRAL EQUATIONS ON THE DISK

We propose preconditioners for the Helmholtz scattering problems by a planar, disk-shaped screen in R-3. Those preconditioners are approximations of the square-roots of some partial differential operators acting on the screen. Their matrix-vector products involve only a few sparse system resolu- tions and can thus be evaluated cheaply in the context of iterative methods. For the Laplace equation (i.e. for the wavenumber k = 0) with Dirichlet condition on the disk and on regular meshes, we prove that the preconditioned linear system has a bounded condition number uniformly in the mesh size. We further provide numerical evidence indicating that the preconditioners also perform well for large values of k and on locally refined meshes.