Abstract

During the past 50 years, numerical methods for differential eigenvalue problems have developed primarily within the `discretize-then-solve’ framework. These methods compute eigenvalues and eigenfunctions of a differential operator L by discretizing L to obtain a matrix eigenvalue problem. Subsequently, the matrix eigenvalue problem can be solved using standard techniques from numerical linear algebra. Motivated by recent developments in mathematical software for highly adaptive computations with functions, we invert this popular paradigm by designing an eigensolver that manipulates L, rather than intermediate discretizations, and approximates only the functions that L acts on. This `solve-then-discretize’ strategy allows us to leverage spectrally accurate approximation schemes for functions while overcoming fundamental difficulties that have undermined their application to differential eigenvalue problems in the past. The resulting eigensolver is efficient, scalable, and well-conditioned, and is capable of resolving eigenfunctions that exhibit rapid oscillations, layers, and other challenging features. Moreover, the solve-then-discretize approach reveals two key ideas that open the door to computing exotic (yet physically relevant) spectral properties of differential operators that have no finite-dimensional analogues, such as the continuous spectrum and associated spectral measure.