Generalized Yee methods: Scalable symplectic finite element Maxwell solvers

TL;DR

This paper introduces generalized Yee methods (GYMs), a class of structure-preserving finite element Maxwell solvers that extend Yee's method to unstructured meshes, higher-order accuracy, and symplectic particle coupling, while maintaining scalability and long-term accuracy.

Contribution

It demonstrates that Yee's method is a special case of GYMs, develops a novel sparsification strategy SPAI-OP, and extends GYMs to structure-preserving electromagnetic particle-in-cell methods.

Findings

GYMs retain symplecticity under sparse approximations.

SPAI-OP concentrates accuracy at prescribed wave modes.

GYMs enable unstructured meshes and higher-order accuracy.

Abstract

Yee's finite-difference method preserves two crucial properties of Maxwell's equations -- locality and symplecticity -- and thereby enjoys two computational advantages: scalability on high-performance architectures and long-time numerical accuracy. In this work, we show that Yee's method is a special case of a class of structure-preserving finite element methods -- termed generalized Yee methods (GYMs) -- that are designed to retain both crucial properties. GYMs are built from de Rham-conforming finite elements and achieve locality through sparse mass matrices and their sparse approximate inverses (SPAIs). We prove that the symplectic structure of GYMs is invariant under such sparse approximations, freeing the choice of sparsification strategy. We introduce a novel sparsification strategy, SPAI-OP, which concentrates accuracy at prescribed wave modes by operator probing. We further extend GYMs to structure-preserving electromagnetic particle-in-cell (PIC) methods, whose symplecticity over particle trajectories requires the smooth fields afforded by higher-order finite elements. GYMs therefore retain the computational virtues of Yee's method while enabling unstructured meshes, higher-order accuracy, spectral adaptivity, and symplectic particle coupling.