Convergence Analysis of Planewave Expansion Methods for 2D Schrödinger Operators with Discontinuous Periodic Potentials

Abstract

In this paper we consider the problem of computing the spectrum of a two-dimensional Schrödinger operator with discontinuous, periodic potential in two dimensions using Fourier (or planewave expansion) methods. Problems of this kind are currently of great interest in the design of new optical devices to determine band gaps and to compute localized modes in photonic crystal materials. Although Fourier methods may not be every applied mathematician's first choice for this problem because of the discontinuities in the potential, we will show here that, even though (as expected) the convergence is not exponential, the method has several desirable features that make it competitive with other discretization techniques, such as finite element methods, both with respect to implementation and convergence properties. In particular, we will prove that simple preconditioners for the system matrix are optimal leading to a computational complexity of $\mathcal{O}(N\log N)$ in the number of planewaves N (using the fast Fourier transform). Moreover, we derive sharp error estimates that show that the method is essentially third order in the eigenvalues and of order $\frac{3}{2}$ in the eigenfunctions in the $H^1$-norm and $\frac{5}{2}$ in the $L^2$-norm. To improve the planewave expansion method in the case of discontinuous potentials, it has been proposed in the physics literature to replace the discontinuous potential with an effective potential that is smooth, despite the additional error this incurs. We will here answer the question whether this smoothing is worth it. In fact, our convergence analysis of the modified method provides an optimal choice for the smoothing parameter, but it also shows that the overall rate of convergence is no faster than before and so smoothing does not seem to be worth it. All the theoretical results are confirmed in our numerical experiments.