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Unconditionally Optimal Error Analysis of Crank–Nicolson Galerkin FEMs for a Strongly Nonlinear Parabolic System

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Abstract

In this paper, we present unconditionally optimal error estimates of linearized Crank–Nicolson Galerkin finite element methods for a strongly nonlinear parabolic system in \(\mathbb {R}^d\ (d=2,3)\). However, all previous works required certain time-step conditions that were dependent on the spatial mesh size. In order to overcome several entitative difficulties caused by the strong nonlinearity of the system, the proof takes two steps. First, by using a temporal-spatial error splitting argument and a new technique, optimal \(L^2\) error estimates of the numerical schemes can be obtained under the condition \(\tau \ge h\), where \(\tau \) denotes the time-step size and h is the spatial mesh size. Second, we obtain the boundedness of numerical solutions by mathematical induction and inverse inequality when \(\tau \le h\). Then, optimal \(L^2\) and \(H^1\) error estimates are proved in a different way for such case. Numerical results are given to illustrate our theoretical analyses.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, China

    Dongfang Li

  2. Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan, 430074, China

    Dongfang Li

  3. Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong

    Dongfang Li

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

    Jilu Wang

Authors
  1. Dongfang Li
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  2. Jilu Wang
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Correspondence to Jilu Wang.

Additional information

The work of Dongfang Li was supported by a grant from the National Science Foundation of China (Grant No. 11571128), China Postdoctoral Science Foundation (Grant No. 2016M602273) and the Research Grants Council of the Hong Kong Special Administrative Region (Project No. CityU 102613).

The work of Jilu Wang was supported in part by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 102613).

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Li, D., Wang, J. Unconditionally Optimal Error Analysis of Crank–Nicolson Galerkin FEMs for a Strongly Nonlinear Parabolic System. J Sci Comput 72, 892–915 (2017). https://doi.org/10.1007/s10915-017-0381-3

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  • DOI: https://doi.org/10.1007/s10915-017-0381-3

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