In many full-electron models used for electronic structure calculations, the singularities of the potential at the nuclei give rise to wavefunctions with cusp-like singularities. Those solutions are thus not regular in the classical Sobolev sense, but belong to a class of countably normed Sobolev spaces (Babuška-Kondratěv spaces) [1]. Traditional finite element or spectral methods do not perform well in this setting, since their rate of convergence is bounded by the regularity of the solution in (possibly weighted) Sobolev spaces.
In this talk, we present the application of an hP discontinuous Galerkin finite element method to the approximation of a nonlinear Schrödinger equation. hP methods combine spatial refinements in low regularity regions with spectral refinements in high regularity regions, and provide an exponentially convergent approximation to functions in Babuška-Kondratěv spaces [2].
In particular, we give bounds that prove the required regularity of the eigenfunctions and demonstrate exponential convergence of the numerical solution via a priori estimates on the eigenfunction and eigenvalue errors. Furthermore, we show a strategy for the a priori optimization of the approximation space based on asymptotic features of the solutions near the nuclei. Both the theoretical convergence analysis and the a priori space optimization are corroborated by numerical results. This is joint work with Yvon Maday (UPMC, Paris).
[1] H. J.
Flad, R. Schneider, B.-W. Schulze. Asymptotic regularity of solutions
to Hartree-Fock equations with Coulomb potential. Math. Methods
Appl. Sci., 31(18):2172–2201, 2008.
[2] D. Schötzau,
C. Schwab, T. P. Wihler. hp-DGFEM
for Second Order Elliptic Problems in Polyhedra II:
Exponential Convergence. SIAM J. Numer. Anal., 51(4):2005–2035,
2013.