Xiaoying Dai

Convergence and optimal complexity of adaptive finite element eigenvalue computations

In this paper, an adaptive finite element method for elliptic eigenvalue problems is studied. Both uniform convergence and optimal complexity of the adaptive finite element eigenvalue approximation are proved. The analysis is based on a certain relationship between the finite element eigenvalue approximation and the associated finite element boundary value approximation which is also established in the paper.

Three-Scale Finite Element Discretizations for Quantum Eigenvalue Problems

Based on globally and locally coupled discretizations, some three-scale finite element schemes are proposed in this paper for a class of quantum eigenvalue problems. It is shown that the solution of a quantum eigenvalue problem on a fine grid may be reduced to the solution of an eigenvalue problem on a relatively coarse grid, and the solutions of linear algebraic systems on a globally mesoscopic grid and the locally fine grid, and the resulting solution is still very satisfactory.

Finite Volume Discretizations for Eigenvalue Problems with Applications to Electronic Structure Calculations

To introduce the finite volume method to electronic structure calculations, we study a symmetric finite volume scheme for a class of linear eigenvalue problems and present a priori error analysis of the finite volume eigenpair approximations. Based on finite volume-finite element coupled discretizations, in particular, we design several higher order approximate schemes. We also demonstrate a series of numerical experiments in electronic structure calculations that illustrate the effectiveness of our finite volume discretization approaches.

A parallel orbital-updating based plane-wave basis method for electronic structure calculations

Abstract Motivated by the recently proposed parallel orbital-updating approach in real space method [1] , 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.

A Conjugate Gradient Method for Electronic Structure Calculations

In this paper, we study a conjugate gradient method for electronic structure calculations. We propose a Hessian based step size strategy, which together with three orthogonality approaches yields three algorithms for computing the ground state energy of atomic and molecular systems. Under some mild assumptions, we prove that our algorithms converge locally. It is shown by our numerical experiments that the conjugate gradient method is efficient.