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.

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.

Reçu le :
Accepté le :
DOI : 10.1051/m2an/2017058
Classification : 65F10, 65N30, 65N55
Mots clés : HDG, error estimate, superconvergence, Schrödinger equations
Xiong, Chunguang 1 ; Luo, Fusheng 1 ; Ma, Xiuling 1

1
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     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},
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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. http://archive.numdam.org/articles/10.1051/m2an/2017058/

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