In this paper, we study an adaptive planewave method for multiple eigenvalues of second-order elliptic partial equations. Inspired by the technique for the adaptive finite element analysis, we prove that the adaptive planewave method has the linear convergence rate and optimal complexity.
In this paper, we propose and analyze some practical Newton methods for electronic structure calculations. We show the convergence and the local quadratic convergence rate for the Newton method when the Newton search directions are well-obtained. In
particular, we investigate some basic implementation issues in determining the search directions and step sizes which ensures the convergence of the subproblem at each iteration and accelerates the algorithm, respectively. It is shown by our numerical experiments that our Newton methods perform better than the existing conjugate gradient method, and the Newton method with the adaptive step size strategy is even more efficient.
Motivated by the recently proposed parallel orbital-updating approach in real space method, we propose a parallel orbital-updating based plane-wave basis method for electronic structure calculations, for solving the corresponding eigenvalue problems.
In addition, we propose two new modified parallel orbital-updating methods. Compared to the traditional plane-wave methods, our methods allow for two-level parallelization, which is particularly interesting for large scale parallelization. Numerical experiments show that these new methods are more reliable and efficient for large scale calculations on modern supercomputers
