Journal of Applied Mathematics
Volume 2012 (2012), Article ID 938727, 26 pages
http://dx.doi.org/10.1155/2012/938727
Research Article

Adaptive Double-Diffusion Model and Comparison to a Highly Heterogeneous Micro-Model

Viviane Klein1 and Malgorzata Peszynska2

1Campus Araranguá, Universidade Federal de Santa Catarina, Rua Pedro João Pereira, 150, 88900-000 Araranguá, SC, Brazil
2Department of Mathematics, Oregon State University, 368 Kidder Hall, Corvallis, Oregon 97331-4605, USA

Received 21 March 2012; Accepted 13 April 2012

Academic Editor: Khalida Inayat Noor

Copyright © 2012 Viviane Klein and Malgorzata Peszynska. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Double-diffusion model is used to simulate slightly compressible fluid flow in periodic porous media as a macro-model in place of the original highly heterogeneous micro-model. In this paper, we formulate an adaptive two-grid numerical finite element discretization of the double-diffusion system and perform a comparison between the micro- and macro-model. Our numerical results show that the micro-model solutions appear to converge to the macro-model linearly with the parameter ε of periodic geometry. For the two-grid discretization, the a priori and a posteriori error estimates are proved, and we show how to adapt the grid for each component independently.