A variational $\boldsymbol{H}({\rm div})$ finite-element discretization approach for perfect incompressible fluids

Abstract

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 |$\boldsymbol{H}({\rm div})$| 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 Guzmán 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.