The gap-tooth scheme for homogenization problems

Giovanni Samaey, Dirk Roose, Ioannis G. Kevrekidis

Research output: Contribution to journalArticlepeer-review

60 Scopus citations


An important class of problems exhibits smooth behavior in space and time on a macroscopic scale, while only a microscopic evolution law is known. For such time-dependent multiscale problems, an "equation-free framework" has been proposed, of which the gap-tooth scheme is an essential component. The gap-tooth scheme is designed to approximate a time-stepper for an unavailable macroscopic equation in a macroscopic domain; it uses appropriately initialized simulations of the available microscopic model in a number of small boxes, which cover only a fraction of the domain. We analyze the convergence of this scheme for a parabolic homogenization problem with nonlinear reaction. In this case, the microscopic model is a partial differential equation with rapidly oscillating coefficients, while the unknown macroscopic model is approximated by the homogenized equation. We show that our method approximates a finite difference scheme of arbitrary (even) order for the homogenized equation when we appropriately constrain the microscopic problem in the boxes. We illustrate this theoretical result with numerical tests on several model problems. We also demonstrate that it is possible to obtain a convergent scheme without constraining the microscopic code, by introducing buffer regions around the computational boxes.

Original languageEnglish (US)
Pages (from-to)278-306
Number of pages29
JournalMultiscale Modeling and Simulation
Issue number1
StatePublished - 2005

All Science Journal Classification (ASJC) codes

  • General Chemistry
  • Modeling and Simulation
  • Ecological Modeling
  • General Physics and Astronomy
  • Computer Science Applications


  • Coarse integration
  • Equation-free framework
  • Gap-tooth scheme
  • Homogenization


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