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

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 to the finite element setting. The resulting algorithm coincides with the energyconserving 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.One of the most recent approaches is the one associated to the school of Brenier [8,9], see, e.g., the discretisations proposed in [17,28]. Such methods are based on the reformulation of the variational principle underlying the incompressible Euler equations into an optimal transport problem. This leads to the definition of the notion of generalised incompressible flow, in which the motion of the fluid particles is defined in a probabilistic sense. The methods in [17,28] aim at producing approximations of the flow map in this setting. In particular, they construct minimizing geodesics between discrete measure preserving maps transporting the particle mass distributions. Moreover, they establish convergence to classical solutions of the Euler equations.In this paper we follow a different route, which is inspired by the work of Pavlov et al. [30,31]. In this case, the discrete flow maps are discretised by considering their action on piecewise constant scalar functions intended as right composition. However, such maps are not constructed explicitly and the final algorithm is fully Eulerian, i.e. one approximates the evolution of the velocity field at fixed positions in space, without following each fluid particle. One key ingredient to achieve this is the identification of vector fields with discrete Lie derivatives, i.e. advection operators, also acting on piecewise constant scalar functions. Owing to this identification, one can reduce the variational principle governing the dynamics from the space of discrete flow maps to the one of discrete velocity fields, i.e. from a Lagrangian (material) description to an Eulerian (spatial) one. The final algorithm can be interpreted as an advection problem for the velocity one-form by means of another discrete Lie derivative operator, defined consistently with the Discrete Exterior Calculus (DEC) formalism [15]. Desbrun et al. [14] extended this promising idea to generate a variational numerical scheme for rotating stratified fluid models. Both the algorithms presented in [31] and [14] ca...

Scale‐selective dissipation in energy‐conserving finite‐element schemes for two‐dimensional turbulence

“…We analyzed the Lie derivative finite‐element discretization of the incompressible Euler equations introduced in Natale and Cotter (2017) and the SUPG discretization of the vorticity advection equation, in terms of energy and enstrophy tendencies in a forced turbulence test case. The main points of this article can be summarized as follows.…”

“…The main points of this article can be summarized as follows. The Lie derivative upwind discretization in Natale and Cotter (2017) can be reformulated to model energy backscatter from small to large scales in a simple way, by an appropriate decomposition of the velocity finite‐element space. The SUPG scheme, on the other hand, can be interpreted as a way to model enstrophy transfer towards unresolved scales. Energy conservation in both schemes does not rely on an a priori choice of a vorticity perturbation pattern, as is usually the case for methods based on energy fixers (Thuburn et al , 2014).…”

“…Alternatively, using standard vector calculus identities, we can also write it as where , and ||·|| 2 denotes the Euclidean norm. We refer to as the Lie derivative formulation, as the advection term in this formulation may be interpreted as the Lie derivative of the velocity one‐form (Natale and Cotter, 2017). Then, the general form of the mixed finite‐element discretization of the system in requires us to find such that for all , where can represent either P or p .…”

“…However, interpreting the stabilization term as we did for the flux form discretization is not straightforward. Natale and Cotter (2017) noticed that, for smooth advecting velocity, such a discretization coincides with the Eulerian Lie derivative discretization proposed in Heumann and Hiptmair (2011) for the linear advection–diffusion problem. In section 3, we elaborate on this observation and give a multiscale interpretation to the stabilization term.…”

“…Numerical tests show that both the flux form and the Lie derivative scheme converge with order r + 1, in terms of velocity L 2 error, when standard upwinding is employed, i.e. α = 1; see Guzmán et al (2015) and Natale and Cotter (2017). A priori convergence estimates with suboptimal convergence rate r can be found in the same references.…”