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Crouzeix-Raviart MsFEM with Bubble Functions for Diffusion and Advection-Diffusion in Perforated Media

Published online by Cambridge University Press:  30 April 2015

Pierre Degond
Affiliation:
Université de Toulouse, UPS, INSA, UT1, UTM; Institut de Mathématiques de Toulouse; F-31062 Toulouse, France CNRS; Institut de Mathématiques de Toulouse UMR 5219; F-31062 Toulouse, France
Alexei Lozinski
Affiliation:
Laboratoire de Mathématiques, UMR CNRS 6623, Université de Franche-Comté, 25030 Besançn Cedex, France
Bagus Putra Muljadi*
Affiliation:
Université de Toulouse, UPS, INSA, UT1, UTM; Institut de Mathématiques de Toulouse; F-31062 Toulouse, France CNRS; Institut de Mathématiques de Toulouse UMR 5219; F-31062 Toulouse, France Department of Earth Science and Engineering, Imperial College London, London SW7 2BP, United Kingdom
Jacek Narski
Affiliation:
Université de Toulouse, UPS, INSA, UT1, UTM; Institut de Mathématiques de Toulouse; F-31062 Toulouse, France CNRS; Institut de Mathématiques de Toulouse UMR 5219; F-31062 Toulouse, France
*
*Corresponding author. Email addresses: b.muljadi@imperial.ac.uk (B. P. Muljadi), alexei.lozinski@univ-fcomte.fr (A. Lozinski), pierre.degond@math.univ-toulouse.fr (P. Degond), jacek.narski@math.univ-toulouse.fr (J. Narski)
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Abstract

The adaptation of Crouzeix-Raviart finite element in the context of multi-scale finite element method (MsFEM) is studied and implemented on diffusion and advection-diffusion problems in perforated media. It is known that the approximation of boundary condition on coarse element edges when computing the multiscale basis functions critically influences the eventual accuracy of any MsFEM approaches. The weakly enforced continuity of Crouzeix-Raviart function space across element edges leads to a natural boundary condition for the multiscale basis functions which relaxes the sensitivity of our method to complex patterns of perforations. Another ingredient to our method is the application of bubble functions which is shown to be instrumental in maintaining high accuracy amid dense perforations. Additionally, the application of penalization method makes it possible to avoid complex unstructured domain and allows extensive use of simpler Cartesian meshes.

Type
Research Article
Copyright
Copyright © Global-Science Press 2015 

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