  Providing evidence of finite-time singularities of the incompressible Euler
equations in three space dimensions is still an unsolved problem. Likewise, the
zeroth law of turbulence has not been proven to date by numerical experiments.
We address this issue by high-resolution numerical simulations of the inviscid
three-dimensional Taylor-Green vortex problem using a novel high-order
discontinuous Galerkin discretization approach. Our main finding is that the
kinetic energy evolution does not tend towards exact energy conservation for
increasing spatial resolution of the numerical scheme, but instead converges to
a solution with nonzero kinetic energy dissipation rate. This implies an energy
dissipation anomaly in the absense of viscous dissipation according to
Onsager's conjecture, and serves as an indication of finite-time singularities
in incompressible inviscid flows. We demonstrate convergence to a dissipative
solution for the three-dimensional inviscid Taylor-Green problem with a
measured relative $L_2$-error of $0.27 \%$ for the temporal evolution of the
kinetic energy and $3.52 \%$ for the kinetic energy dissipation rate.
