@Article{IJNAM-14-2,
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 = {2017},
volume = {14},
number = {2},
pages = {283--305},
abstract = {<p style="text-align: justify;">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.</p>},
issn = {2617-8710},
doi = {https://doi.org/2017-IJNAM-421},
url = {https://global-sci.com/article/83168/error-analysis-of-a-finite-difference-scheme-for-the-epitaxial-thin-film-model-with-slope-selection-with-an-improved-convergence-constant}
}