The gaptooth scheme for homogenization problems
Abstract
An important class of problems exhibits smooth behaviour in space and time on a macroscopic scale, while only a microscopic evolution law is known. For such timedependent multiscale problems, an ``equationfree framework'' has been proposed, of which the gaptooth scheme is an essential component. The gaptooth scheme is designed to approximate a timestepper 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.
 Publication:

arXiv eprints
 Pub Date:
 December 2003
 DOI:
 10.48550/arXiv.physics/0312004
 arXiv:
 arXiv:physics/0312004
 Bibcode:
 2003physics..12004S
 Keywords:

 Physics  Computational Physics
 EPrint:
 32 pages