We consider the application of finite element exterior calculus (FEEC) methods to a class of canonical Hamiltonian PDE systems involving differential forms. Solutions to these systems satisfy a local multisymplectic conservation law, which generalizes the more familiar symplectic conservation law for Hamiltonian systems of ODEs, and which is connected with physically-important reciprocity phenomena, such as Lorentz reciprocity in electromagnetics. We characterize hybrid FEEC methods whose numerical traces satisfy a version of the multisymplectic conservation law, and we apply this characterization to several specific classes of FEEC methods, including conforming Arnold-Falk-Winther-type methods and various hybridizable discontinuous Galerkin (HDG) methods. Interestingly, the HDG-type and other nonconforming methods are shown, in general, to be multisymplectic in a stronger sense than the conforming FEEC methods. This substantially generalizes previous work of McLachlan and Stern [Found. Comput. Math., 20 (2020), pp. 35-69] on the more restricted class of canonical Hamiltonian PDEs in the de Donder-Weyl "grad-div" form.

翻译：我们考虑将有限元外微积分（FEEC）方法应用于一类涉及微分形式的典型哈密顿偏微分方程组。这些系统的解满足局部多辛守恒律，该守恒律推广了常微分方程哈密顿系统中更熟悉的辛守恒律，并与物理上重要的互易现象（如电磁学中的洛伦兹互易性）相关联。我们刻画了混合FEEC方法，其数值迹满足多辛守恒律的某一种形式，并将该刻画应用于几类具体的FEEC方法，包括相容的Arnold-Falk-Winther型方法以及多种可杂交间断伽辽金（HDG）方法。有趣的是，一般而言，HDG型方法及其他非相容方法在比相容FEEC方法更强的意义上具有多辛性。这实质性地推广了McLachlan与Stern[Found. Comput. Math., 20 (2020), pp. 35-69]关于更受限的de Donder-Weyl“梯度-散度”形式的典型哈密顿偏微分方程的先前工作。