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Wang, Jianyun; Tian, Zhikun

Superconvergence of finite element approximations of the two-dimensional cubic nonlinear Schrödinger equation. (English) Zbl 1499.65686

Adv. Appl. Math. Mech. 14, No. 3, 652-665 (2022).
Summary: The superconvergence of a two-dimensional time-independent nonlinear Schrödinger equation are analyzed with the rectangular Lagrange type finite element of order \(k\). Firstly, the error estimate and superclose property are given in \(H^1\)-norm with order \(\mathcal{O}(h^{k+1})\) between the finite element solution \(u_h\) and the interpolation function \(u_I\) by use of the elliptic projection operator. Then, the global superconvergence is obtained by the interpolation post-processing technique. In addition, some numerical examples with the order \(k = 1\) and \(k = 2\) are provided to demonstrate the theoretical analysis.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)

Keywords:

superconvergence; nonlinear Schrödinger equation; finite element method; elliptic projection
PDFBibTeX XMLCite
\textit{J. Wang} and \textit{Z. Tian}, Adv. Appl. Math. Mech. 14, No. 3, 652--665 (2022; Zbl 1499.65686)
Full Text: DOI

References:

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