Spectral Exterior Calculus and Its Implementation

Abstract

Preserving geometric, topological and algebraic structures at play in partial differential equations has proven to be a fruitful guiding principle for computational methods in a variety of scientific fields. However, structure-preserving numerical methods have traditionally used spaces of piecewise polynomial basis functions with local support to interpolate differential forms. When solutions are known to be smooth, a spectral treatment is often preferred instead as it brings exponential convergence. While recent works have established spectral variants of discrete exterior calculus, no existing approach offers the full breadth of exterior calculus operators and a clear distinction between vectors and covectors. We present such a unified approach to spectral exterior calculus (SPEX) and provide detail on its implementation. Notably, our approach leverages Poincare duality through the use of a primal grid and its dual (with a natural handling of boundaries to facilitate the treatment of boundary conditions), and uses a twin representation of differential forms as both integrated and pointwise values. Through its reliance on the fast Fourier transform, the resulting framework enables computations in arbitrary dimensions that are both efficient and have excellent convergence properties.