Volume 14, Issue 2
Error Analysis of a Finite Difference Scheme for the Epitaxial Thin Film Model with Slope Selection with an Improved Convergence Constant

Z.-H. Qiao, C. Wang, S. M. Wise & Z.-R. Zhang

Int. J. Numer. Anal. Mod., 14 (2017), pp. 283-305.

Published online: 2016-05

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  • Abstract

In this paper we present an improved error analysis for a finite difference scheme for solving the 1-D epitaxial thin film model with slope selection. The unique solvability and unconditional energy stability are assured by the convex nature of the splitting scheme. A uniform-in-time $H^m$ bound of the numerical solution is acquired through Sobolev estimates at a discrete level. It is observed that a standard error estimate, based on the discrete Gronwall inequality, leads to a convergence constant of the form exp($CT\varepsilon^{-m}$), where $m$ is a positive integer, and $\varepsilon$ is the corner rounding width, which is much smaller than the domain size. To improve this error estimate, we employ a spectrum estimate for the linearized operator associated with the 1-D slope selection (SS) gradient flow. With the help of the aforementioned linearized spectrum estimate, we are able to derive a convergence analysis for the finite difference scheme, in which the convergence constant depends on $\varepsilon^{-1}$ only in a polynomial order, rather than exponential.

  • Keywords

Epitaxial thin film growth, finite difference, convex splitting, uniform-in-time $H^m$ stability, linearized spectrum estimate, discrete Gronwall inequality.

  • AMS Subject Headings

35K30, 65M06, 65M12, 65T40

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-14-283, author = {}, title = {Error Analysis of a Finite Difference Scheme for the Epitaxial Thin Film Model with Slope Selection with an Improved Convergence Constant}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2016}, volume = {14}, number = {2}, pages = {283--305}, abstract = {

In this paper we present an improved error analysis for a finite difference scheme for solving the 1-D epitaxial thin film model with slope selection. The unique solvability and unconditional energy stability are assured by the convex nature of the splitting scheme. A uniform-in-time $H^m$ bound of the numerical solution is acquired through Sobolev estimates at a discrete level. It is observed that a standard error estimate, based on the discrete Gronwall inequality, leads to a convergence constant of the form exp($CT\varepsilon^{-m}$), where $m$ is a positive integer, and $\varepsilon$ is the corner rounding width, which is much smaller than the domain size. To improve this error estimate, we employ a spectrum estimate for the linearized operator associated with the 1-D slope selection (SS) gradient flow. With the help of the aforementioned linearized spectrum estimate, we are able to derive a convergence analysis for the finite difference scheme, in which the convergence constant depends on $\varepsilon^{-1}$ only in a polynomial order, rather than exponential.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/421.html} }
TY - JOUR T1 - Error Analysis of a Finite Difference Scheme for the Epitaxial Thin Film Model with Slope Selection with an Improved Convergence Constant JO - International Journal of Numerical Analysis and Modeling VL - 2 SP - 283 EP - 305 PY - 2016 DA - 2016/05 SN - 14 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/421.html KW - Epitaxial thin film growth, finite difference, convex splitting, uniform-in-time $H^m$ stability, linearized spectrum estimate, discrete Gronwall inequality. AB -

In this paper we present an improved error analysis for a finite difference scheme for solving the 1-D epitaxial thin film model with slope selection. The unique solvability and unconditional energy stability are assured by the convex nature of the splitting scheme. A uniform-in-time $H^m$ bound of the numerical solution is acquired through Sobolev estimates at a discrete level. It is observed that a standard error estimate, based on the discrete Gronwall inequality, leads to a convergence constant of the form exp($CT\varepsilon^{-m}$), where $m$ is a positive integer, and $\varepsilon$ is the corner rounding width, which is much smaller than the domain size. To improve this error estimate, we employ a spectrum estimate for the linearized operator associated with the 1-D slope selection (SS) gradient flow. With the help of the aforementioned linearized spectrum estimate, we are able to derive a convergence analysis for the finite difference scheme, in which the convergence constant depends on $\varepsilon^{-1}$ only in a polynomial order, rather than exponential.

Z.-H. Qiao, C. Wang, S. M. Wise & Z.-R. Zhang. (1970). Error Analysis of a Finite Difference Scheme for the Epitaxial Thin Film Model with Slope Selection with an Improved Convergence Constant. International Journal of Numerical Analysis and Modeling. 14 (2). 283-305. doi:
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