A variational H(div) finite element discretisation approach for perfect incompressible fluids
For computational fluid dynamics, this provides a geometrically consistent finite element method for ideal fluids, though it builds on existing schemes and is domain-specific.
The paper extends a structure-preserving discretisation of the incompressible Euler equations to finite elements, yielding an energy-conserving scheme that also satisfies a discrete Kelvin's circulation theorem. An upwind-stabilised version dissipates enstrophy while preserving energy and Kelvin's theorem, with proven error estimates and numerical validation.
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 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-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.