SIMULTANEOUS MULTIGRID TECHNIQUES FOR NONLINEAR EIGENVALUE PROBLEMS -SOLUTIONS OF THE NONLINEAR SCHRODINGER-POISSON EIGENVALUE PROBLEM IN 2 AND 3 DIMENSIONS

Algorithms for nonlinear eigenvalue problems (EP's) often require solv ing 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 eigenval ues 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. T his paper presents MG techniques and a MG algorithm for nonlinear Schr odinger-Poisson EP's. The algorithm overcomes the above mentioned diff iculties combining the following techniques: a MG simultaneous treatme nt of the eigenvectors and nonlinearity, and with the global constrain ts; MG stable subspace continuation techniques for the treatment of no nlinearity; and a MG projection coupled with backrotations for separat ion of solutions. These techniques keep the solutions in an appropriat e region, where the algorithm converges fast, and reduce the large num ber of self-consistent iterations to only a few or one MG simultaneous iteration. The MG projection makes it possible to efficiently overcom e difficulties related to clusters of close and equal eigenvalues. Com putational examples for the nonlinear Schrodinger-Poisson EP in two an d three dimensions, presenting special computational difficulties, tha t are due to the nonlinearity and to the equal and closely clustered e igenvalues are demonstrated. For these cases, the algorithm requires O (qN) operations for the calculation of q eigenvectors of size N and fo r the corresponding eigenvalues. One MG simultaneous cycle per fine le vel was performed. The total computational cost is equivalent to only a few Gauss-Seidel relaxations per eigenvector. An asymptotic converge nce rate of 0.15 per MG cycle is attained.