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Superconvergence analysis of Crank‐Nicolson Galerkin FEMs for a generalized nonlinear Schrödinger equation
Author(s) -
Zhang Houchao,
Wang Junjun
Publication year - 2018
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22230
Subject(s) - superconvergence , mathematics , crank–nicolson method , norm (philosophy) , nonlinear system , lemma (botany) , finite element method , galerkin method , mathematical analysis , element (criminal law) , numerical analysis , ecology , physics , poaceae , quantum mechanics , biology , political science , law , thermodynamics
The purpose of this article is to apply E Q 1 rotnonconforming finite element(FE) to solve a generalized nonlinear Schrödinger equation. First, a new important property of E Q 1 rotnonconforming FE (see (2.3) of Lemma 2 below) is proved by use of BHX lemma and the integral identities techniques. Second, a linearized Crank‐Nicolson fully discrete scheme is constructed and the superclose error estimate of order O ( h 2 + τ 2 ) for original variable u in broken H 1 ‐norm is also derived by using the properties of E Q 1 rotelement and the splitting argument for nonlinear terms, while previous works always only obtain convergent error estimates with this element. Furthermore, the global superconvergence is arrived at by the interpolated postprocessing technique. Finally, two numerical experiments are provided to confirm the theoretical analysis. Here, h is the subdivision parameter and τ is the time step.