A variational H(div) finite element discretisation approach for perfect incompressible fluids

TL;DR

This paper introduces a novel finite element discretisation for the incompressible Euler equations that preserves geometric structure, energy, and circulation properties, with added stability and error analysis.

Contribution

It extends structure-preserving discretisation to finite elements using a variational approach, including an upwind-stabilised scheme with proven error estimates.

Findings

The scheme conserves energy and Kelvin's circulation theorem.

The upwind-stabilised version dissipates enstrophy while maintaining energy.

Numerical tests validate the theoretical properties and error estimates.

Abstract

We propose a finite element discretisation 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 discretisation of Pavlov et al. (2009) to the finite element setting. The resulting algorithm coincides with the energy-conserving scheme presented in 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-stabilised version of the scheme which dissipates enstrophy whilst 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.