The J-method for the Gross-Pitaevskii eigenvalue problem
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This paper studies the J-method of [E. Jarlebring, S. Kvaal, W. Michiels. SIAM J. Sci. Comput. 36-4:A1978-A2001, 2014] for nonlinear eigenvector problems in a general Hilbert space framework. This is the basis for variational discretization techniques and a mesh-independent numerical analysis. A simple modification of the method mimics an energy-decreasing discrete gradient flow. In the case of the Gross-Pitaevskii eigenvalue problem we prove global convergence towards an eigenfunction. More importantly, a local linear rate of convergence is established. This quantitative convergence analysis is closely connected to the J-method's unique feature of sensitivity with respect to spectral shifts. Contrary to classical gradient flows this allows both the selective approximation of excited states as well as the amplification of convergence beyond linear rates in the spirit of the Rayleigh quotient iteration for linear eigenvalue problems. These advantageous convergence properties are demonstrated in a series of numerical experiments involving exponentially localized states under disorder potentials and vortex lattices in rotating traps.
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