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This is a correction to: IMA Journal of Numerical Analysis, Volume 38, Issue 3, 17 July 2018, Pages 1388–1419, https://doi.org/10.1093/imanum/drx033

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.

We review the role of dual pairs in mechanics and use them to derive particle-like solutions to regularized incompressible fluid systems. In our case we have a dual pair resulting from the action of diffeomorphisms on point particles (essentially by moving the points). We then augment our dual pair by considering the action of diffeomorphisms on Taylor series, also known as jets. The augmented weak dual pairs induce a hierarchy of particle-like solutions and conservation laws with particles carrying a copy of a jet group. We call these augmented particles jetlets. The jet groups serve as finite-dimensional models of the diffeomorphism group itself, and so the jetlet particles serve as a finite-dimensional model of the self-similarity exhibited by ideal incompressible fluids. The conservation law associated to jetlet solutions is shown to be a shadow of Kelvin’s circulation theorem. Finally, we study the dynamics of infinite time particle mergers. We prove that two merging particles at the zeroth level in the hierarchy yield dynamics which asymptotically approach that of a single particle in the first level in the hierarchy. This merging behavior is then verified numerically as well as the exchange of angular momentum which must occur during a near collision of two particles. The resulting particle-like solutions suggest a new class of meshless methods which work in dimensions n≥2n≥2 and which exhibit a shadow of Kelvin’s circulation theorem. More broadly, this provides one of the first finite-dimensional models of self-similarity in ideal fluids.