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| Newton linearization has been successfully used in [1] for solving fluid dynamics and heat transfer (Navier-Stokes-Boussinesq) equations with the advantage to accelerate computations due to its rapid quadratic convergence. Sobolev gradient methods proved very effective in minimizing constrained Schr\"{o}dinger type energy functionals, as the Gross-Pitaevskii energy [2].
Since the Newton method can be viewed as a steepest descent gradient method with variable metric, we can combine the Newton method with the Sobolev gradient method to solve unconstrained or constrained minimization problems. We illustrate the new method by computing various cases from fluid mechanics (phase-change materials) and condensed matter physics (Bose-Einstein condensates).
[1] I. Danaila, R. Moglan, F. Hecht, S. Le Masson, A Newton method with adaptive finite elements for solving phase-change problems with natural convection, submitted, 2013.
[2] 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. |
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