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Unconditional Superconvergence Analysis of a Crank–Nicolson Galerkin FEM for Nonlinear Schrödinger Equation

Abstract

A linearized Crank–Nicolson Galerkin finite element method with bilinear element for nonlinear Schrödinger equation is studied. By splitting the error into two parts which are called the temporal error and the spatial error, the unconditional superconvergence result is deduced. On one hand, the regularity for a time-discrete system is presented based on the proof of the temporal error. On the other hand, the classical Ritz projection is applied to get the spatial error with order \(O(h^2)\) in \(L^2\)-norm, which plays an important role in getting rid of the restriction of \(\tau \). Then the superclose estimates of order \(O(h^2+\tau ^2)\) in \(H^1\)-norm is arrived at based on the relationship between the Ritz projection and the interpolated operator. At the same time, global superconvergence property is arrived at by the interpolated postprocessing technique. At last, three numerical examples are provided to confirm the theoretical analysis. Here, h is the subdivision parameter and \(\tau \) is the time step.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 11271340).

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Correspondence to Dongyang Shi.

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Shi, D., Wang, J. Unconditional Superconvergence Analysis of a Crank–Nicolson Galerkin FEM for Nonlinear Schrödinger Equation. J Sci Comput 72, 1093–1118 (2017). https://doi.org/10.1007/s10915-017-0390-2

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Keywords

  • Unconditional superconvergence results
  • NLSE
  • Linearized C–N Galerkin FEM
  • Temporal and spatial errors
  • Ritz projection and interpolated operators

Mathematics Subject Classification

  • 65N15
  • 65N30