No CrossRef data available.
Article contents
Crouzeix-Raviart MsFEM with Bubble Functions for Diffusion and Advection-Diffusion in Perforated Media
Published online by Cambridge University Press: 30 April 2015
Abstract
Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.
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.
Keywords
Information
- Type
- Research Article
- Information
- Copyright
- Copyright © Global-Science Press 2015
References
[1]Angot, P., Bruneau, C. H., and Fabrie, P.. A penalization method to take into account obstacles in incompressible viscous flows. Numer. Math., (81):497–520, 1999.CrossRefGoogle Scholar
[2]Babuška, I., Banerjee, U., and Osborn, J.E.. Survey of meshless and generalized finite element methods: A unified approach. Acta Numerica, pages 1–125, 2003.CrossRefGoogle Scholar
[3]Bourgeat, A.. Homogenized behaviour of two-phase flows in naturally fractured reservoirs with uniform fractures distribution. Comp. Meth. Appl. Mech. Eng., 47:205–215, 1984.CrossRefGoogle Scholar
[4]Carballal Perdiz, L.. Etude d’une méthodologie multiéchelles appliquée à différents problèmes en milieu continu et discret. PhD thesis, Institut de Mathématiques de Toulouse, 2011.Google Scholar
[5]Chu, J., Efendiev, Y., Ginting, V., and Hou, T.Y.. Flow based oversampling technique for multiscale finite element methods. Advances in Water Resources, 31(4):599 – 608, 2008.CrossRefGoogle Scholar
[6]Cioranescu, D., Donato, P., and Zaki, R.. Periodic unfolding and robin problems in perforated domains. C.R. Acad. Sci. Paris, 342:469–474, 2006.CrossRefGoogle Scholar
[7]Cioranescu, D. and Murat, F.. A strange term coming from nowhere, in topics in the mathematical modelling of composite materials. In Progress in Nonlinear Differential Equations and their applications, volume 31, pages 45–93. Birkhauser, 1997.Google Scholar
[8]Crouzeix, M. and Raviart, P. A.. Conforming and nonconforming finite element methods for solving the stationary stokes equations i. RAIRO, 7(3):33–75, 1973.Google Scholar
[9]Deng, W., Yun, X., and Xie, C.. Convergence analysis of the multiscale method for a class of convectiondiffusion equations with highly oscillating coefficients. Applied Numerical Mathematics, 59(7):1549–1567, 2009.CrossRefGoogle Scholar
[10]Dorobantu, M. and Engquist, B.. Wavelet-based numerical homogenization. SIAM J. Numer. Anal., (35):540–559, 1998.CrossRefGoogle Scholar
[11]Efendiev, Y. and Hou, T.. Multiscale finite element methods for porous media flows and their applications. Applied Numerical Mathematics, 57(57):577 – 596, 2007.CrossRefGoogle Scholar
[12]Efendiev, Y. and Hou, T. Y.. Multiscale finite element method, theory and applications. Surveys and tutorials in the applied mathematical sciences. Springer, New York, 2009.Google Scholar
[13]Efendiev, Y., Galvis, J., Li, G., and Presho, M.. Generalized multiscale finite element methods. oversampling strategies. arXiv:1304.4888, 2013.CrossRefGoogle Scholar
[14]Henning, P. and Ohlberger, M.. The heterogeneous multiscale finite element method for elliptic homogenization problems in perforated domains. Numer. Math., 113(4):601–629, 2009.CrossRefGoogle Scholar
[15]Henning, P. and Peterseim, D.. Oversampling for the multiscale finite element method. arXiv:1211.5954, 2012.CrossRefGoogle Scholar
[16]Hornung, U.. Homogenization and Porous Media, Interdisciplinary Applied Mathematics, volume 6. Springer, 1997.CrossRefGoogle Scholar
[17]Hou, T. Y. and Wu, X. H.. A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys., 134(1):169–189, 1997.CrossRefGoogle Scholar
[18]Kevrekidis, I.G., Gear, C.W., Hyman, J.M., Kevrekidis, P.G., Runborg, O., and Theodoropoulos, C.. Equation-free, coarse-grained multiscale computation: enabling microscopic simulators to perform system-level analysis. Commun. Math. Sci., 1(4):715–762, 2003.Google Scholar
[19]Le Bris, C., Legoll, F., and Lozinski, A.. Msfem à la crouzeix-raviart for highly oscillatory elliptic problems. Chinese Annals of Mathematics, Series B, 34(1):113–138, 2013.CrossRefGoogle Scholar
[20]Le Bris, C., Legoll, F., and Lozinski, A.. An msfem type approach for perforated domains. arXiv:1307.0876, 2013.CrossRefGoogle Scholar
[21]Lions, J. L.. Asymptotic expansions in perforated media with a periodic structure. Rocky Mountain J. of Maths, 10(1):125–140, 1980.Google Scholar
[22]Nolen, J., Papanicolaou, G., and Pironneau, O.. A framework for adaptive multiscale method for elliptic problems. SIAM MMS, 7:171–196, 2008.CrossRefGoogle Scholar
[23]Weinan, E. and Engquist, B.. The heterogeneous multi-scale methods. Comm. Math. Sci., 1(1):87–133, 2003.Google Scholar
You have Access- 20
- Cited by