Error analysis of a finite difference scheme for the epitaxial thin film model with slope selection with an improved convergence constant

Zhonghua Qiao, Cheng Wang, Steven M. Wise, Zhengru Zhang

Research output: Journal article publicationJournal articleAcademic researchpeer-review

15 Citations (Scopus)

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 Hmbound 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(CTε−m), where m is a positive integer, and ε 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 ε−1only in a polynomial order, rather than exponential.
Original languageEnglish
Pages (from-to)283-305
Number of pages23
JournalInternational Journal of Numerical Analysis and Modeling
Volume14
Issue number2
Publication statusPublished - 1 Jan 2017

Keywords

  • Convex splitting
  • Discrete Gronwall inequality
  • Epitaxial thin film growth
  • Finite difference
  • Linearized spectrum estimate
  • Uniform-in-time H stability m

ASJC Scopus subject areas

  • Numerical Analysis

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