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A Second Order Energy Stable Linear Scheme for a Thin Film Model Without Slope Selection

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Abstract

In this paper we present a second order accurate, energy stable numerical scheme for the epitaxial thin film model without slope selection, with a mixed finite element approximation in space. In particular, an explicit treatment of the nonlinear term, \(\frac{\nabla u}{1+|\nabla u|^2}\), greatly simplifies the computational effort; only one linear equation with constant coefficients needs to be solved at each time step. Meanwhile, a second order Douglas–Dupont regularization term, \(A\tau \varDelta ^2 ( u^{n+1} - u^n)\), is added in the numerical scheme, so that an unconditional long time energy stability is assured. In turn, we perform an \(\ell ^\infty (0,T; L^2)\) convergence analysis for the proposed scheme, with an \(O (\tau ^2 + h^q)\) error estimate derived. In addition, an optimal convergence analysis is provided for the nonlinear term using \(Q_q\) finite elements, which shows that the spatial convergence order can be improved to \(q+1\) on regular rectangular mesh. A few numerical experiments are presented, which confirms the efficiency and accuracy of the proposed second order numerical scheme.

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Acknowledgements

This work is supported in part by the Grants NSFC 11671098, 11331004, 91630309, a 111 Project B08018 (W. Chen), NSF DMS-1418689 (C. Wang), and Grant 2017110715 by Shanghai University of Finance and Economics (Y. Yan). C. Wang also thanks Shanghai Center for Mathematical Sciences and Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, for support during his visit. We thank the anonymous reviewers for their careful reading of our manuscript and their many insightful comments and suggestions, which lead to substantial improvements of this paper.

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Authors and Affiliations

  1. School of Mathematical Sciences, Fudan University, Shanghai, 200433, China

    Weijia Li

  2. School of Mathematical Sciences and Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Shanghai, 200433, China

    Wenbin Chen

  3. Mathematics Department, University of Massachusetts, North Dartmouth, MA, 02747, USA

    Cheng Wang

  4. School of Mathematics, Shanghai University of Finance and Economics, Shanghai, 200433, China

    Yue Yan

  5. Department of Chemical and Petroleum Engineering, Schulich School of Engineering, University of Calgary, Calgary, AB, T2N 1N4, Canada

    Ruijian He

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  1. Weijia Li
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  2. Wenbin Chen
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  3. Cheng Wang
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Correspondence to Yue Yan.

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Li, W., Chen, W., Wang, C. et al. A Second Order Energy Stable Linear Scheme for a Thin Film Model Without Slope Selection. J Sci Comput 76, 1905–1937 (2018). https://doi.org/10.1007/s10915-018-0693-y

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  • DOI: https://doi.org/10.1007/s10915-018-0693-y

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