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Unconditional Optimal Error Estimates of BDF–Galerkin FEMs for Nonlinear Thermistor Equations

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Abstract

In this paper we study linearized backward differential formula (BDF) type schemes with Galerkin finite element approximations for the time-dependent nonlinear thermistor equations. Optimal \(L^2\) error estimates for the proposed schemes are proved unconditionally. The proof consists of two steps. First, the boundedness of the numerical solution in certain strong norms is obtained by a temporal-spatial error splitting argument. Second, a traditional approach is used to provide an optimal \(L^2\) error estimate for \(r\)-th order FEMs \((r \ge 1)\). Numerical experiments in both two and three dimensional spaces are conducted to confirm our theoretical analysis and show the high order accuracy and unconditional stability (convergence) of the linearized BDF–Galerkin FEMs.

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Acknowledgments

The author would like to thank Professor Weiwei Sun for valuable suggestions and many constructive discussions. The author is grateful to two anonymous referees for their valuable comments on earlier versions of this paper; especially for one referee bringing our attention to the regularity assumption of the exact solutions.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, People’s Republic of China

    Huadong Gao

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  1. Huadong Gao
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Correspondence to Huadong Gao.

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H. Gao: The work of the author was supported in part by Fundamental Research Funds for the Central Universities, HUST, China, under Grant No. 2014QNRC025, No. 2015QN132 and a Grant from the Research Grants Council of the Hong Kong Special Administrative Region, China, Project No. CityU 11302514.

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Gao, H. Unconditional Optimal Error Estimates of BDF–Galerkin FEMs for Nonlinear Thermistor Equations. J Sci Comput 66, 504–527 (2016). https://doi.org/10.1007/s10915-015-0032-5

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  • DOI: https://doi.org/10.1007/s10915-015-0032-5

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