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Unconditional stability and convergence of Crank–Nicolson Galerkin FEMs for a nonlinear Schrödinger–Helmholtz system

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Abstract

The paper is concerned with the unconditional stability and optimal \(L^2\) error estimates of linearized Crank–Nicolson Galerkin FEMs for a nonlinear Schrödinger–Helmholtz system in \({\mathbb {R}}^d\) (\(d=2,3\)). By introducing a corresponding time-discrete system, we separate the error into two parts, i.e., the temporal error and the spatial error. Since the latter is \(\tau \)-independent, the uniform boundedness of numerical solutions in \(L^{\infty }\)-norm follows an inverse inequality immediately without any restrictions on time stepsize. Then, optimal error estimates are obtained in a routine way. Numerical examples in both two and three dimensional spaces are given to illustrate our theoretical results.

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Acknowledgements

The author would like to thank Professor Weiwei Sun for the valuable discussions. The author would also like to thank the anonymous referees for their suggestions and comments, which helped to improve the quality of the paper.

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  1. Department of Scientific Computing, Florida State University, Tallahassee, FL, 32306, USA

    Jilu Wang

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  1. Jilu Wang
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Correspondence to Jilu Wang.

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The work of the author was supported in part by the USA National Science Foundation Grant DMS-1315259, the USA Air Force Office of Scientific Research Grant FA9550-15-1-0001, and a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 11300517).

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Wang, J. Unconditional stability and convergence of Crank–Nicolson Galerkin FEMs for a nonlinear Schrödinger–Helmholtz system. Numer. Math. 139, 479–503 (2018). https://doi.org/10.1007/s00211-017-0944-0

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  • DOI: https://doi.org/10.1007/s00211-017-0944-0

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