  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.
