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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.
Xiong, Chunguang 1 ; Luo, Fusheng 1 ; Ma, Xiuling 1
@article{M2AN_2018__52_2_751_0,
author = {Xiong, Chunguang and Luo, Fusheng and Ma, Xiuling},
title = {Uniform in time error analysis of {HDG} approximation for {Schr\"odinger} equation based on {HDG} projection},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {751--772},
publisher = {EDP-Sciences},
volume = {52},
number = {2},
year = {2018},
doi = {10.1051/m2an/2017058},
mrnumber = {3834442},
zbl = {1416.65365},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2017058/}
}
TY - JOUR AU - Xiong, Chunguang AU - Luo, Fusheng AU - Ma, Xiuling TI - Uniform in time error analysis of HDG approximation for Schrödinger equation based on HDG projection JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 751 EP - 772 VL - 52 IS - 2 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an/2017058/ DO - 10.1051/m2an/2017058 LA - en ID - M2AN_2018__52_2_751_0 ER -
%0 Journal Article %A Xiong, Chunguang %A Luo, Fusheng %A Ma, Xiuling %T Uniform in time error analysis of HDG approximation for Schrödinger equation based on HDG projection %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 751-772 %V 52 %N 2 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/m2an/2017058/ %R 10.1051/m2an/2017058 %G en %F M2AN_2018__52_2_751_0
Xiong, Chunguang; Luo, Fusheng; Ma, Xiuling. Uniform in time error analysis of HDG approximation for Schrödinger equation based on HDG projection. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 2, pp. 751-772. doi: 10.1051/m2an/2017058
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