Pollution and accuracy of solutions of the Helmholtz equation: A novel perspective from the eigenvalues

In researching the Helmholtz equation, the focus has either been on the accuracy of the numerical solution (pollution) or the acceleration of the convergence of a preconditioned Krylov-based solver (scalability). While it is widely recognized that the convergence properties can be investigated by studying the eigenvalues, information from the eigenvalues is not used in studying the numerical dispersion which drives the pollution error. Our aim is to bring the topics of accuracy and scalability together for the first time; instead of approaching the pollution error in the conventional sense of being the result of a discrepancy between the exact and numerical wavenumber, we show that the dispersion which drives the pollution error can also be decomposed in terms of the eigenvectors and eigenvalues. Using these novel insights, we construct sharper upper bounds for the total error independent of the grid resolution. While the pollution error can be minimized in one-dimension by introducing a dispersion correction, the latter is not possible in higher dimensions, even for very simple model problems. For our model problem, a correction on the eigenvalues enables us to remove the pollution error and study it in full detail, both in one- and two-dimensions.