Finite element exterior calculus for time-dependent Hamiltonian partial differential equations

TL;DR

This paper develops a class of structure-preserving numerical methods for time-dependent Hamiltonian PDEs by combining finite element exterior calculus with symplectic integrators, ensuring local multisymplectic conservation.

Contribution

It introduces a novel framework that generalizes symplectic integration to PDEs using FEEC, capturing finer Hamiltonian structures than previous methods.

Findings

Methods satisfy local multisymplectic conservation law.

Application to semilinear Hodge wave equation demonstrates effectiveness.

Particular focus on conforming FEEC and HDG methods.

Abstract

The success of symplectic integrators for Hamiltonian ODEs has led to a decades-long program of research seeking analogously structure-preserving numerical methods for Hamiltonian PDEs. In this paper, we construct a large class of such methods by combining finite element exterior calculus (FEEC) for spatial semidiscretization with symplectic integrators for time discretization. The resulting methods satisfy a local multisymplectic conservation law in space and time, which generalizes the symplectic conservation law of Hamiltonian ODEs, and which carries finer information about Hamiltonian structure than other approaches based on global function spaces. We give particular attention to conforming FEEC methods and hybridizable discontinuous Galerkin (HDG) methods. The theory and methods are illustrated by application to the semilinear Hodge wave equation.