  We propose a finite element discretisation approach for the incompressible
Euler equations which mimics their geometric structure and their variational
derivation. In particular, we derive a finite element method that arises from a
nonholonomic variational principle and an appropriately defined Lagrangian,
where finite element H(div) vector fields are identified with advection
operators; this is the first successful extension of the structure-preserving
discretisation of Pavlov et al. (2009) to the finite element setting. The
resulting algorithm coincides with the energy-conserving scheme presented in
Guzm\'an et al. (2016). Through the variational derivation, we discover that it
also satisfies a discrete analogous of Kelvin's circulation theorem. Further,
we propose an upwind-stabilised version of the scheme which dissipates
enstrophy whilst preserving energy conservation and the discrete Kelvin's
theorem. We prove error estimates for this version of the scheme, and we study
its behaviour through numerical tests.
