We propose a finite-element discretization 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 discretization of
Pavlov
et al.
(
2009
) to the finite-element setting. The resulting
algorithm coincides with the energy-conserving scheme proposed by
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-stabilized version of the scheme that dissipates enstrophy while 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.