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A mixed finite element method for thin film epitaxy

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Abstract

We present a mixed finite element method for the thin film epitaxy problem. Comparing to the primal formulation which requires \(C^2\) elements in the discretization, the mixed formulation only needs to use \(C^1\) elements, by introducing proper dual variables. The dual variable in our method is defined naturally from the nonlinear term in the equation, and its accurate approximation will be essential for understanding the long-time effect of the nonlinear term. For time-discretization, we use a backward-Euler semi-implicit scheme, which involves a convex–concave decomposition of the nonlinear term. The scheme is proved to be unconditionally stable and its convergence rate is analyzed.

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Acknowledgments

Chen was supported by the 111 project, Key Project National Science Foundation of China (91130004) and the Natural Science Foundation of China (11171077). He also thanks Jianguo Liu in Duke University and Xiaoming Wang in Florida State University for the fruitful discussions. Wang thanks the Key Laboratory of Mathematics for Nonlinear Sciences (EYH1140070), Fudan University, for the support during her visit. The authors are also grateful to the anonymous referees for their helpful comments and suggestions which greatly improved the quality of this paper.

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Authors and Affiliations

  1. School of Mathematical Sciences, Fudan University, Shanghai, China

    Wenbin Chen

  2. Department of Mathematics, Oklahoma State University, Stillwater, OK, USA

    Yanqiu Wang

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  1. Wenbin Chen
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Correspondence to Yanqiu Wang.

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Chen, W., Wang, Y. A mixed finite element method for thin film epitaxy. Numer. Math. 122, 771–793 (2012). https://doi.org/10.1007/s00211-012-0473-9

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  • DOI: https://doi.org/10.1007/s00211-012-0473-9

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