  We introduce generalized Galerkin variational integrators, which are a
natural generalization of discrete variational mechanics, whereby the discrete
action, as opposed to the discrete Lagrangian, is the fundamental object. This
is achieved by approximating the action integral with appropriate choices of a
finite-dimensional function space that approximate sections of the
configuration bundle and numerical quadrature to approximate the integral. We
discuss how this general framework allows us to recover higher-order Galerkin
variational integrators, asynchronous variational integrators, and
symplectic-energy-momentum integrators. In addition, we will consider function
spaces that are not parameterized by field values evaluated at nodal points,
which allows the construction of Lie group, multiscale, and pseudospectral
variational integrators. The construction of pseudospectral variational
integrators is illustrated by applying it to the (linear) Schrodinger equation.
G-invariant discrete Lagrangians are constructed in the context of Lie group
methods through the use of natural charts and interpolation at the level of the
Lie algebra. The reduction of these G-invariant Lagrangians yield a
higher-order analogue of discrete Euler-Poincare reduction. By considering
nonlinear approximation spaces, spatio-temporally adaptive variational
integrators can be introduced as well.
