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Optimal Convergence Analysis of a Second Order Scheme for a Thin Film Model Without Slope Selection

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Abstract

In Li et al. (J Sci Comput 76(3):1905–1937, 2018), a temporal second-order mixed finite element scheme has been proposed for the thin film epitaxial growth model without slope selection. Using the super-convergence theory in a regular rectangular mesh, the authors of Li et al. (2018) proved an optimal \(O(h^{q+1}+\tau ^2)\) convergence. However, in a quasi-uniform triangulation mesh setting, only a sub-optimal convergence rate \(O(h^q+\tau ^2)\) is proved, while numerical results indicated an optimal \(O(h^{q+1}+\tau ^2)\) convergence when the exact solution has \(H^{q+1}\) regularity in space. Here h and \(\tau \) are the discretization sizes in space and time, respectively, and \(q\ge 1\) is the degree of the polynomial in the spatial discretization. In this paper, we provide a theoretical proof of the optimal convergence rate. The main difficulty lies in how to treat a nonlinear term \(\frac{\nabla u}{1+|\nabla u|^2}\). We solve this by using a discrete Laplacian operator \(-\varDelta _h\) and some uncommon techniques in the analysis. Numerical results are also presented to demonstrate the \((q+1)\)-order convergence of the spatial approximation.

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Acknowledgements

This work is supported in part by the Grants NSFC 11671098, 11331004, 91630309, a 111 Project B08018 (W. Chen), the Grants NSFC 11671210, 91630201 (Y. Wang), and the Grant 2017110715 (Y. Yan). Y. Yan also thanks the support by Institute of Scientific Computation and Financial Data Analysis, Shanghai University of Finance and Economics.

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Authors and Affiliations

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

    Wenbin Chen

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

    Yichao Zhang & Weijia Li

  3. School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China

    Yanqiu Wang

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

    Yue Yan

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  1. Wenbin Chen
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  2. Yichao Zhang
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  3. Weijia Li
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  4. Yanqiu Wang
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  5. Yue Yan
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Correspondence to Yanqiu Wang.

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Chen, W., Zhang, Y., Li, W. et al. Optimal Convergence Analysis of a Second Order Scheme for a Thin Film Model Without Slope Selection. J Sci Comput 80, 1716–1730 (2019). https://doi.org/10.1007/s10915-019-00999-y

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