Rate of convergence of general phase field equations in strongly heterogeneous media towards their homogenized limit

Markus Schmuck, Serafim Kalliadasis

Research output: Contribution to journalArticle

1 Citation (Scopus)
122 Downloads (Pure)

Abstract

Over the last few decades, phase field equations have found increasing applicability in a wide range of mathematical-scientific fields (e.g., geometric PDEs and mean curvature flow, materials science for the study of phase transitions) but also engineering ones (e.g., as a computational tool in chemical engineering for interfacial flow studies). Here, we focus on phase field equations in strongly heterogeneous materials with perforations such as porous media. To the best of our knowledge, we provide the first derivation of error estimates for fourth order, homogenized, and nonlinear evolution equations. Our fourth order problem induces a slightly lower convergence rate, i.e., ∈1/4, where e denotes the material's specific heterogeneity, than established for second order elliptic problems (e.g., [V. ZHIKOV, Dokl. Math., 73(2006), pp. 96-99, https://doi.org/10.1134/S1064562406010261.]) for the error between the effective macroscopic solution of the (new) upscaled formulation and the solution of the microscopic phase field problem. We hope that our study will motivate new modeling, analytic, and computational perspectives for interfacial transport and phase transformations in strongly heterogeneous environments.

Original languageEnglish
Pages (from-to)1471-1492
Number of pages22
JournalSIAM Journal on Applied Mathematics
Volume77
Issue number4
Early online date24 Aug 2017
DOIs
Publication statusE-pub ahead of print - 24 Aug 2017

Keywords

  • upscaling
  • homogenization
  • free energy
  • phase field transformations
  • error estimates
  • convergence rates

ASJC Scopus subject areas

  • Applied Mathematics

Fingerprint Dive into the research topics of 'Rate of convergence of general phase field equations in strongly heterogeneous media towards their homogenized limit'. Together they form a unique fingerprint.

  • Cite this