Convergence to Equilibrium of a DC Algorithm for an Epitaxial Growth Model

Authors

  • Hamza Khalfi LAMAI Laboratory, Faculty of Science and Technology, Cadi Ayyad University, Marrakesh, Morocco
  • Morgan Pierre Laboratoire de Mathématiques et Applications, Université de Poitiers, CNRS, F-86962 Chasseneuil, France
  • Nour Eddine Alaa LAMAI Laboratory, Faculty of Science and Technology, Cadi Ayyad University, Marrakesh, Morocco
  • Mohammed Guedda LAMFA Laboratory, CNRS UMR 7352, Picardie Jules Verne University, Amiens, France

Keywords:

Thin film epitaxy, DC programming, coarsening dynamics, Lojasiewicz-Simon inequality, epitaxial growth, model without slope selection, Fourier spectral method, convergence to equilibrium, pattern formation.

Abstract

A linear numerical scheme for an epitaxial growth model is analyzed in this work. The considered scheme is already established in the literature using a convexity splitting argument. We show that it can be naturally derived from an optimization viewpoint using a DC (difference of convex functions) programming framework. Moreover, we prove the convergence of the scheme towards equilibrium by means of the Lojasiewicz-Simon inequality. The fully discrete version, based on a Fourier collocation method, is also analyzed. Finally, numerical simulations are carried out to accommodate our analysis.

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Published

2018-11-22

Issue

Vol. 16 No. 3 (2019)

Section

Articles