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

Numerical evidence of anomalous energy dissipation in incompressible  Euler flows: Towards grid-converged results for the inviscid Taylor-Green  problem