Simultaneous multigrid techniques for nonlinear eigenvalue problems: Solutions of the nonlinear Schrödinger-Poisson eigenvalue problem in two and three dimensions

作     者：Sorin Costiner Shlomo Ta’asan 

作者机构：Institute for Computer Applications in Science and Engineering NASA Langley Research Center Hampton Virginia 23681 

出 版 物：《Physical Review E》 (物理学评论E辑：统计、非线性和软体物理学)

年 卷 期：1995年第52卷第1期

页      面：1181-1181页

核心收录：

学科分类：07[理学] 070203[理学-原子与分子物理] 0702[理学-物理学] 

摘      要：Algorithms for nonlinear eigenvalue problems (EP’s) often require solving self-consistently a large number of EP’s. Convergence difficulties may occur if the solution is not sought in an appropriate region, if global constraints have to be satisfied, or if close or equal eigenvalues are present. Multigrid (MG) algorithms for nonlinear problems and for EP’s obtained from discretizations of partial differential EP have often been shown to be more efficient than single level algorithms. This paper presents MG techniques and a MG algorithm for nonlinear Schrödinger Poisson EP’s. The algorithm overcomes the above mentioned difficulties combining the following techniques: a MG simultaneous treatment of the eigenvectors and nonlinearity, and with the global constrains; MG stable subspace continuation techniques for the treatment of nonlinearity; and a MG projection coupled with backrotations for separation of solutions. These techniques keep the solutions in an appropriate region, where the algorithm converges fast, and reduce the large number of self-consistent iterations to only a few or one MG simultaneous iteration. The MG projection makes it possible to efficiently overcome difficulties related to clusters of close and equal eigenvalues. Computational examples for the nonlinear Schrödinger-Poisson EP in two and three dimensions, presenting special computational difficulties that 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 fine 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.