*Shuo Zhang (Institute of Computational Mathematics, Chinese Academy of Sciences)

For practical and theoretical reasons, high-efficiency schemes are of wide interests. The issue of high efficiency means more information obtained and/or less computational cost. The obtainments include a better approximation for source and eigenvalue problems and, e.g., a guaranteed upper or lower bound of the eigenvalue, while the cost includes computational resources and time for computation. In this talk, we would like to present some algorithms which seek to give higher-efficiency performance for eigenvalue problems. The high efficiency comes by different approaches. We will mention multigrid algorithm, high-accuracy discretization schemes and improvement of the estimation of upper/lower bounds, up to the time limitation. Some non-standard phenomena will be presented.