Mathematical and Numerical Methods for Complex Quantum Systems

A posteriori error estimates for discontinuous Galerkin methods using non-polynomial basis functions with applications to Kohn-Sham density functional theory calculation

Lin Lin

University of California, Berkeley


We develop a posteriori error estimates in the discontinuous Galerkin framework for general non-polynomial basis functions. Compared to standard hp-refinement using polynomial basis functions, the main difficulty for obtaining a reliable estimator is that no a priori asymptotic estimate is available for these general basis functions. Therefore important pre-constants in front of the residual and jump terms in standard a posteriori error estimates are not known. To overcome this difficulty, we develop a general strategy to estimate all these constants on the fly. The extra computational cost of the strategy is small, and the strategy by definition provides the sharpest constants within the residual based a posteriori error estimates even for hp-refinement. We demonstrate the practical use of the a posteriori error estimator in performing three-dimensional Kohn-Sham density functional theory calculations for quasi-2D aluminum surfaces and single-layer graphene oxide in water.