Generalizing the Yee Algorithm to Higher Order on an Unstructured Mesh

It is well known that the Yee algorithm satisfies two crucial properties: (i) It faithfully preserves the geometric structure of Maxwell's equations, ensuring its accuracy in long-time numerical simulations; and (ii) its calculations are local and therefore parallelizable, enabling Yee's method to capitalize on the speed and scalability of high-performance computing architecture. In this work, Yee's algorithm is recast in the formalism of finite element exterior calculus and, in contrast with its usual finite-difference interpretation, it is thereby viewed as a low order finite element method with simplified mass matrices. Previous attempts to improve upon Yee's method with finite elements have sacrificed the indispensable computational efficiency afforded by its localness. Here, we leverage the finite element point of view to generalize the Yee algorithm to higher order and general meshes, while developing innovative techniques to maintain its geometric naturalness, physically-motivated localness, and parallel efficiency, nevertheless.