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.
Accepté le :
DOI : 10.1051/m2an/2017058
Mots-clés : HDG, error estimate, superconvergence, Schrödinger equations
@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://archive.numdam.org/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://archive.numdam.org/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://archive.numdam.org/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. http://archive.numdam.org/articles/10.1051/m2an/2017058/
[1] A high order finite element discretization with local absorbing boundary conditions of the linear Schrödinger equation. J. Comput. Phys. 220 (2006) 409–421. | DOI | MR | Zbl
and ,[2] Numerical schemes for simulation of the two-dimensional Schrödinger equation using non-reflecting boundary conditions. Math. Comput. 73 (2004) 1779–1799. | DOI | MR | Zbl
, and ,[3] Analysis of a leap-frog pseudospectral scheme for the Schrödinger equation. J. Comput. Appl. Math. 193 (2006) 65–88. | DOI | MR | Zbl
and ,[4] Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York (1991) 90–132. | MR | Zbl
and ,[5] Uniform-in-time superconvergence of HDG methods for the heat equation. Math. Comput. 81 (2012) 107–129. | DOI | MR | Zbl
and ,[6] Superconvergence of mixed finite element methods for parabolic problems with nonsmooth initial data. Numer. Math. 78 (1998) 495–521. | DOI | MR | Zbl
, and ,[7] Convergence analysis of a finite volume method for Maxwell’s equations in nonhomogeneous media. SIAM J. Numer. Anal. 41 (2003) 37–63. | DOI | MR | Zbl
, and ,[8] A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems. Math. Comput. 77 (2008) 1887–1916. | DOI | MR | Zbl
, and ,[9] A hybridizable and superconvergent discontinuous Galerkin method for biharmonic problems. J. Scient. Comput. 40 (2009) 141–187. | DOI | MR | Zbl
, and ,[10] Superconvergent discontinuous Galerkin methods for second-order elliptic problems. Math. Comput. 78 (2009) 1–24. | DOI | MR | Zbl
, and ,[11] Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 47 (2009) 1319–1365. | DOI | MR | Zbl
, and ,[12] A priori error analysis for HDG methods using extensions from subdomains to achieve boundary conformity. Math. Comput. 83 (2014) 665–699. | DOI | MR | Zbl
, and ,[13] A new multiscale discontinuous Galerkin method for the one-dimensional stationary Schrödinger equation. J. Sci Comput. 66 (2016) 321–345. | DOI | MR | Zbl
, and ,[14] Superconvergence of the mixed finite element approximations of parabolic problems using rectangular finite elements. East-West J. Numer. Math. 1 (1993) 199–212. | MR | Zbl
and ,[15] A generalized discontinuous Galerkin (GDG) method for Schrödinger equations with nonsmooth solutions,J. Comput. Phys. 227 (2008) 2387–2410. | DOI | MR | Zbl
, and ,[16] An analysis of HDG methods for convection-dominated diffusion problems. ESAIM: M2AN 49 (2015) 225–256. | DOI | Numdam | MR | Zbl
, and ,[17] Sharp maximum norm error estimates for general mixed finite element approximations to second order elliptic equations. RAIRO Model. Math. Anal. Numer. 23 (1989) 103–128. | DOI | Numdam | MR | Zbl
and ,[18] A projection-based error analysis of HDG methods. Math. Comput. 79 (2010) 1351–1367. | DOI | MR | Zbl
, and ,[19] A finite difference method for the one-dimensional time-dependent Schrödinger equation on unbounded domain. Comput. Math. Appl., 50 (2005) 1345–1362. | DOI | MR | Zbl
, and ,[20] Quantum Wells, Wires and Dots. John Wiley & Sons, New York (2000).
,[21] Numerical studies on nonlinear Schrödinger equations by spectral collocation method with preconditioning. J. Math. Anal. Appl. 333 (2007) 1119–1127. | DOI | MR | Zbl
and ,[22] Analysis of finite element method for one-dimensional time-dependent Schrödinger equation on unbounded domain. J. Comput. Appl. Math. 220 (2008) 240–256. | DOI | MR | Zbl
and ,[23] Error estimate of fourth-order compact scheme for linear Schrödinger equations. SIAM J. Numer. Anal. 47 (2010) 4381–4401. | DOI | MR | Zbl
, and ,[24] Conservative local discontinuous Galerkin methods for time dependent Schrödinger equation. Int. J. Numer. Anal. Model. 2 (2014) 72–80. | MR | Zbl
, and ,[25] An absolutely stable hp-HDG method for the time-harmonic Maxwell equations with high wave number. Math. Comput. 86 (2017) 1553–1577. | DOI | MR | Zbl
, and ,[26] Hybridizable discontinuous Galerkin methods, in Spectral and High Order Methods for Partial Differential Equations. Springer (2011) 63–84. | DOI | MR | Zbl
, and ,[27] An HDG method for convection diffusion equation. J. Scient. Comput. 66 (2016) 346–357. | DOI | MR | Zbl
and ,[28] A superconvergent HDG methods for the incompressible Navier–Stokes equations on general polyhedral meshes. IMA J. Numer. Anal. 36 (2016) 1–18. | MR
and ,[29] Finite Element and Boundary Element Applications in Quantum Mechanics. Oxford University Press, Oxford (2002). | MR | Zbl
,[30] A mixed finite element method for second order elliptic problems, in Mathematical Aspects of Finite Element Method. Vol. 606 of Lecture Notes in Math., edited by and . Springer-Verlag, New York (1977) 292–315. | DOI | MR | Zbl
and ,[31] From Raviart–Thomas to HDG: a personal voyage. Preprint (2013) | arXiv
,[32] Quantum Mechanics. McGraw-Hill, New York (1968).
,[33] Postprocessing schemes for some mixed finite elements. RAIRO Model. Math. Anal. Numer. 25 (1991) 151–167. | DOI | Numdam | MR | Zbl
,[34] A new error analysis of Crank–Nicolson Galerkin FEMs for a generalized nonlinear Schrödinger equation. J. Sci. Comput. 60 (2014) 390–407. | DOI | MR | Zbl
,Cité par Sources :