Uniform in time error analysis of HDG approximation for Schrödinger equation based on HDG projection

¹ Department of Mathematics, Beijing Institute of Technology, Beijing 100081, PR China
² Third Institute of Oceanography, State Oceanic Administration, No. 178 Daxue Road, Siming District, Xiamen 361005, PR China

^* Corresponding author: xiongchg@bit.edu.cn

Received: 5 March 2016
Accepted: 13 November 2017

Abstract

This paper presents error analysis of hybridizable discontinuous Galerkin (HDG) time-domain method for solving time dependent Schrödinger equations. The numerical trace and numerical flux are constructed to preserve the conservative property for the density of the particle described. We prove that there exist the superconvergence properties of the HDG method, which do hold for second-order elliptic problems, uniformly in time for the semidiscretization by the same method of Schrödinger equations provided that enough regularity is satisfied. Thus, if the approximations are piecewise polynomials of degree r, the approximations to the wave function and the flux converge with order r + 1. The suitably chosen projection of the wave function into a space of lower polynomial degree superconverges with order r + 2 for r ≥ 1 uniformly in time. The application of element-by-element postprocessing of the approximate solution which provides an approximation of the potential convergence with order r + 2 for r ≥ 1 in L² is also uniformly in time.