Convergence of supercell and superspace methods for computing spectra of quasiperiodic operators

We study the convergence of two of the most widely used and intuitive approaches for computing the spectra of differential operators with quasiperiodic coefficients: the supercell method and the superspace method. In both cases, Floquet–Bloch theory for periodic operators can be used to compute approximations to the spectrum. We present error estimates, derive explicit constructions of supercells needed to obtain optimal convergence rates and show how to choose the numerical scheme to reduce spectral pollution in the superspace method. We illustrate our results with examples of Schrödinger and Helmholtz operators.