28.06.2018 17:15 Geneviève Dusson (University of Warwick):
A posteriori error estimates for eigenvalue problems arising from electronic structure calculationsMI 03.08.011 (Boltzmannstr. 3, 85748 Garching)

(joint work with Eric Cances, Yvon Maday, Benjamin Stamm and Martin Vohralík) The determination of the electronic structure of a molecular system usually requires the resolution of a nonlinear eigenvalue problem. To be solved numerically, this problem is first discretized, using for example finite elements or planewaves. Then the discrete nonlinear equations are solved using an iterative algorithm. At this point, a natural question arising is the size of the error between the computed solution and the exact solution of the given model. In this talk, I will first present an a posteriori error estimation for the eigenvectors and eigenvalues of a Schrödinger-type linear eigenvalue problem. The a posteriori bounds are guaranteed, fully computable, and converge with optimal speed to the given exact eigenvalues and eigenvectors of the problem [1]. These bounds are valid under very few assumptions that can be checked numerically. Also, they can be generalized for clusters of eigenvalues. Numerical simulations conirm the efficiency of the bounds. In a second part, I will illustrate how these estimations on linear eigenvalue problems can be used for the error estimation of nonlinear eigenvalue problems appearing in electronic structure calculations [2].

References [1] Eric Cances, Geneviève Dusson, Yvon Maday, Benjamin Stamm, and Martin Vohralík, Guaranteed and robust a posteriori bounds for Laplace eigenvalues and eigenvectors: conforming approximations, SIAM Journal on Numerical Analysis 55 (5), 2228-2254. [2] Eric Cances, Geneviève Dusson, Yvon Maday, Benjamin Stamm, and Martin Vohralí, A perturbation-method-based a posteriori estimator for the planewave discretization of nonlinear Schrödinger equations, Comptes Rendus Mathematique 352 (11), 941-946.