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 parameterεof 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.
Highlights
IntroductionWhen modeling phenomena in highly heterogeneous media, one frequently finds that the coefficients of differential equations describing these phenomena vary by several orders of magnitude between close-by locations
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
When modeling phenomena in highly heterogeneous media, one frequently finds that the coefficients of differential equations describing these phenomena vary by several orders of magnitude between close-by locations
Summary
When modeling phenomena in highly heterogeneous media, one frequently finds that the coefficients of differential equations describing these phenomena vary by several orders of magnitude between close-by locations. Various multiscale modeling techniques have been introduced with the aim to derive, analyze, and approximate micro-models by the macro-models. There are few results devoted to the comparison between the micro- and macro-models for evolution equations and to their adaptive numerical discretizations. We propose a two-grid adaptive finite element approximation of the following multiscale averaged model for 1.1 in highly heterogeneous media such as porous media with fractures φ1. In this paper we extend those to the evolution system 1.2a and 1.2b and present an a posteriori error estimator of residual type based partly on the work in 14–16 for scalar equations. We provide the technical proofs of some of the results derived in this paper
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