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Optimal Error Analysis of Galerkin FEMs for Nonlinear Joule Heating Equations

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Abstract

We study in this paper two linearized backward Euler schemes with Galerkin finite element approximations for the time-dependent nonlinear Joule heating equations. By introducing a time-discrete (elliptic) system as proposed in Li and Sun (Int J Numer Anal Model 10:622–633, 2013; SIAM J Numer Anal (to appear)), we split the error function as the temporal error function plus the spatial error function, and then we present unconditionally optimal error estimates of \(r\)th order Galerkin FEMs (\(1 \le r \le 3\)). Numerical results in two and three dimensional spaces are provided to confirm our theoretical analysis and show the unconditional stability (convergence) of the schemes.

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Acknowledgments

The author would like to thank Professor Weiwei Sun for valuable suggestions and many constructive discussions. The work of the author was supported in part by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 102712).

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

    Huadong Gao

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Gao, H. Optimal Error Analysis of Galerkin FEMs for Nonlinear Joule Heating Equations. J Sci Comput 58, 627–647 (2014). https://doi.org/10.1007/s10915-013-9746-4

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