Local strategies for improving the conditioning of the plane-wave Ultra-Weak Variational Formulation

•Numerical solution of the wave equation in the frequency domain.•Use of wave-based methods to avoid dispersion.•Cheap treatment of plane-wave discretization ill-conditioning.•Robust and efficient numerical solution of the Helmholtz equation.•Methods avoiding spurious resonances induced by the use of direct solvers. Element-wise techniques based on SVD or QR Decomposition completely get rid of the usual ill-conditioning inherent to the plane-wave discretizations of the Ultra-Weak Variational Formulation (UWVF) for the Helmholtz equation. Associated preconditioning strategies lead to very low condition numbers of the corresponding linear system matrices, without any limitation on the number of plane waves per element. In addition, some of these procedures have the advantage of considerably reducing the size of the final system to be solved without altering the accuracy of the numerical solution.