NESTED ITERATIONS FOR SYMMETRICAL EIGENPROBLEMS

The evaluation of the leftmost eigenspectrum of large sparse symmetric matrices is of great interest in many physical and engineering applic ations. Recently an efficient iterative method called DACG (deflation- accelerated conjugate gradient) based on the conjugate gradient optimi zation of successive deflated Rayleigh quotients, was developed. In th e present paper the authors study the possibility of increasing the ef ficiency of this scheme by means of a nested iteration (NI) technique that acts on nested finite element grids and calculates an improved in itial guess to be used by the DACG procedure. The new NI-DACG method i s tested on two sample problems of large size, which make use of neste d regular and irregular finite element grids. The results obtained in the calculation of the 40 leftmost eigenpairs for both problems show t hat the computational efficiency of DACG is increased by a factor of f ive for the regular mesh and two for the irregular one. They also show the limitation of simple (low-order) interpolation schemes for the in tergrid transfer in improving the initial eigenvector estimates, and c all for more research in this area.