Steepest descent methods using Sobolev gradients proved very effective to solve minimisation problems in different application fields [1].
We present two original methods to solve minimisation problems using Sobolev gradients.
The first problem concerns the reconstruction of the velocity field in a fluid flow dominated by a large scale vortex ring structure.
We reconstruct the vorticity distribution inside the axisymmetric vortex ring from some incomplete and possibly noisy measurements of the surrounding velocity field. The numerical approach inspired from shape optimisation techniques is described in detail in [2].

The second problem is the minimisation of the constrained Gross-Pitaevskii type energy functional describing superfluid Bose-Einstein condensates. We present the new Sobolev gradient method suggested in [3] to efficiently compute stationary states with quantized vortices and used in [4] to simulate rotating Bose-Einstein condensates. Extensions of the numerical system to simulate Quantum Turbulence will be also shortly presented [5].

Both numerical algorithms were implemented using a finite-element method and programmed using the free software FreeFem++ [6], an easy-to-use and highly adaptive software offering many advantages for the implementation of complex algorithms: syntax close to the mathematical formulation, advanced automatic mesh generator, mesh adaptation, automatic interpolation, interface with state-of-the-art numerical libraries (PETSC, UMFPACK, SUPERLU, MUMPS, METIS, IPOPT, etc).
We illustrate the suggested new numerical methods by computing various cases from fluid mechanics (vortex rings)
and condensed matter physics (Bose-Einstein condensates with quantized vortices).

[1] J. W. Neuberger, Sobolev Gradients and Differential Equations, Springer, 2010.

[2] I. Danaila and B. Protas, Optimal reconstruction of inviscid vortices, Proceedings of the Royal Society A, 471: 20150323, 2015.

[3] I. Danaila, P. Kazemi, A new Sobolev gradient method for direct minimization of
the Gross-Pitaevskii energy with rotation. SIAM J. of Scientific Computing,
32:2447-2467, 2010.

[4] G. Vergez, I. Danaila, S. Auliac, F.Hecht, A finite-element toolbox for the stationary Gross-Pitaevskii equation with rotation, Computer Physics Communications, 2016.

[5] M. Kobayashi, Ph. Parnaudeau, F. Luddens, C. Lothodé, L. Danaila, M. Brachet and I. Danaila
Quantum turbulence simulations using the Gross-Pitaevskii equation: high-performance computing and new numerical benchmarks,
Computer Physics Communications, 258, p. 107579(1-26), 2023.

[6] www.freefem.org.