  Variational integrators have traditionally been constructed from the
perspective of Lagrangian mechanics, but there have been recent efforts to
adopt discrete variational approaches to the symplectic discretization of
Hamiltonian mechanics using Hamiltonian variational integrators. In this paper,
we will extend these results to the setting of Hamiltonian multisymplectic
field theories. We demonstrate that one can use the notion of Type II
generating functionals for Hamiltonian partial differential equations as the
basis for systematically constructing Galerkin Hamiltonian variational
integrators that automatically satisfy a discrete multisymplectic conservation
law, and establish a discrete Noether's theorem for discretizations that are
invariant under a Lie group action on the discrete dual jet bundle. In
addition, we demonstrate that for spacetime tensor product discretizations, one
can recover the multisymplectic integrators of Bridges and Reich, and show that
a variational multisymplectic discretization of a Hamiltonian multisymplectic
field theory using spacetime tensor product Runge--Kutta discretizations is
well-defined if and only if the partitioned Runge--Kutta methods are symplectic
in space and time.
