Multigrid Techniques for Nonlinear Eigenvalue Problems; Solutions of a Nonlinear Schrodinger Eigenvalue Problem in 2D and 3D

Algorithms for nonlinear eigenvalue problems (EP), often require solving selfconsistently a large number of EP. Convergence di culties may occur if the solution is not sought in a right neighborhood; if global constraints have to be satis ed; and if close or equal eigenvalues are present. Multigrid (MG) algorithms for nonlinear problems and for EP obtained from discretizations of partial di erential EP, have often shown to be more e cient than single level algorithms. This paper presents MG techniques for nonlinear EP and emphasizes an MG algorithm for a nonlinear Schrodinger EP. The algorithm overcomes the mentioned di culties combining the following techniques: an MG projection coupled with backrotations for separation of solutions and treatment of di culties related to clusters of close and equal eigenvalues; MG subspace continuation techniques for the treatment of the nonlinearity; an MG simultaneous treatment of the eigenvectors at the same time with the nonlinearity and with the global constraints. The simultaneous MG techniques reduce the large number of selfconsistent iterations to only a few or one MG simultaneous iteration and keep the solutions in a right neighborhood where the algorithm converges fast. Computational examples for the nonlinear Schrodinger EP in 2D and 3D, presenting special computational di culties, which are due to the nonlinearity and to the equal and closely clustered eigenvalues, are demonstrated. For these cases, the algorithm requires O(qN ) operations for the calculation of q eigenvectors of size N and for the corresponding eigenvalues. One MG simultaneous cycle per ne level was performed. The total computational cost is equivalent to only a few Gauss-Seidel relaxations per eigenvector. An asymptotic convergence rate of 0.15 per MG cycle is attained. This research was made possible in part by funds granted to Shlomo Ta'asan, a fellowship program sponsored by the Charles H. Revson Foundation. Both authors were supported in part by the National Aeronautics and Space Administration under NASA Contract No. NAS1-19480 while the authors were in residence at the Institute for Computer Applications in Science and Engineering, (ICASE), Mail Stop 132C, NASA Langley Research Center, Hampton, Virginia, 23681, USA.