  Compatible discretizations, such as finite element exterior calculus, provide
a discretization framework that respect the cohomological structure of the de
Rham complex, which can be used to systematically construct stable mixed finite
element methods. Multisymplectic variational integrators are a class of
geometric numerical integrators for Lagrangian and Hamiltonian field theories,
and they yield methods that preserve the multisymplectic structure and
momentum-conservation properties of the continuous system. In this paper, we
investigate the synthesis of these two approaches, by constructing
discretization of the variational principle for Lagrangian field theories
utilizing structure-preserving finite element projections. In our
investigation, compatible discretization by cochain projections plays a pivotal
role in the preservation of the variational structure at the discrete level,
allowing the discrete variational structure to essentially be the restriction
of the continuum variational structure to a finite-dimensional subspace. The
preservation of the variational structure at the discrete level will allow us
to construct a discrete Cartan form, which encodes the variational structure of
the discrete theory, and subsequently, we utilize the discrete Cartan form to
naturally state discrete analogues of Noether's theorem and multisymplecticity,
which generalize those introduced in the discrete Lagrangian variational
framework by Marsden et al. [29]. We will study both covariant spacetime
discretization and canonical spatial semi-discretization, and subsequently
relate the two in the case of spacetime tensor product finite element spaces.
