October  2016, 9(5): 1299-1326. doi: 10.3934/dcdss.2016052

A multiscale finite element method for Neumann problems in porous microstructures

1. 

University of Nottingham, Mathematical Sciences, University Park, Nottingham, NG7 2RD, United Kingdom

2. 

Fraunhofer ITWM, Fraunhofer-Platz 1, 67663 Kaiserslautern, Germany

Received  February 2015 Revised  January 2016 Published  October 2016

In this paper we develop and analyze a Multiscale Finite Element Method (MsFEM) for problems in porous microstructures. By solving local problems throughout the domain we are able to construct a multiscale basis that can be computed in parallel and used on the coarse-grid. Since we are concerned with solving Neumann problems, the spaces of interest are conforming spaces as opposed to recent work for the Dirichlet problem in porous domains that utilizes a non-conforming framework. The periodic perforated homogenization of the problem is presented along with corrector and boundary correction estimates. These periodic estimates are then used to analyze the error in the method with respect to scale and coarse-grid size. An MsFEM error similar to the case of oscillatory coefficients is proven. A critical technical issue is the estimation of Poincaré constants in perforated domains. This issue is also addressed for a few interesting examples. Finally, numerical examples are presented to confirm our error analysis. This is done in the setting of coarse-grids not intersecting and intersecting the microstructure in the setting of isolated perforations.
Citation: Donald L. Brown, Vasilena Taralova. A multiscale finite element method for Neumann problems in porous microstructures. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1299-1326. doi: 10.3934/dcdss.2016052
References:
[1]

A. Abdulle, W. E, B. Engquist and E. Vanden-Eijnden, The heterogeneous multiscale method,, Acta Numer., 21 (2012), 1. Google Scholar

[2]

S. C. Brenner and R. S. Scott, The Mathematical Theory of Finite Element Methods, volume 15 of Texts in Applied Mathematics,, Springer, (2008). doi: 10.1007/978-0-387-75934-0. Google Scholar

[3]

C. L. Bris, F. Legoll and A. Lozinski, An MsFEM type approach for perforated domains,, Multiscale Model. Simul., 12 (2014), 1046. doi: 10.1137/130927826. Google Scholar

[4]

D. L. Brown and D. Peterseim, A Multiscale Method for Porous Microstructures,, Multiscale Model. Simul., 14 (2016), 1123. doi: 10.1137/140995210. Google Scholar

[5]

G. A. Chechkin, A. L. Piatniski and A. S. Shamev, Homogenization: Methods and Applications, volume 234 of Translations of Mathematical Monographs,, American Mathematical Society, (2007). Google Scholar

[6]

D. Cioranescu, A. Damlamian and G. Griso, The periodic unfolding method in homogenization,, SIAM Journal on Mathematical Analysis, 40 (2008), 1585. doi: 10.1137/080713148. Google Scholar

[7]

D. Cioranescu, A. Damlamian, G. Griso and D. Onofrei, The periodic unfolding method for perforated domains and neumann sieve models,, Journal de Mathématiques Pures et Appliquées, 89 (2008), 248. doi: 10.1016/j.matpur.2007.12.008. Google Scholar

[8]

W. E and B. Engquist, The heterogeneous multiscale methods,, Commun. Math. Sci., 1 (2003), 87. Google Scholar

[9]

Y. Efendiev, T. Hou and V. Ginting, Multiscale finite element methods for nonlinear problems and their applications,, Commun. Math. Sci., 2 (2004), 553. Google Scholar

[10]

Y. R. Efendiev and X. Wu, Multiscale finite element for problems with highly oscillatory coefficients,, Numer. Math., 90 (2002), 459. doi: 10.1007/s002110100274. Google Scholar

[11]

Y. R. Efendiev, T. Y. Hou and X.-H. Wu, Convergence of a nonconforming multiscale finite element method,, SIAM Journal on Numerical Analysis, 37 (2000), 888. doi: 10.1137/S0036142997330329. Google Scholar

[12]

G. Griso, Error estimate and unfolding for periodic homogenization,, C. R. Math. Acad. Sci. Paris, 335 (2002), 333. doi: 10.1016/S1631-073X(02)02477-9. Google Scholar

[13]

P. Henning, P. Morgenstern and D. Peterseim, Multiscale Partition of Unity,, In M. Griebel and M. A. Schweitzer, (2014). Google Scholar

[14]

P. Henning and M. Ohlberger, The heterogeneous multiscale finite element method for elliptic homogenization problems in perforated domains,, Numer. Math, 113 (2009), 601. doi: 10.1007/s00211-009-0244-4. Google Scholar

[15]

J. S. Hesthaven, S. Zhang and X. Zhu, High-order multiscale finite element method for elliptic problems,, Multiscale Modeling & Simulation, 12 (2014), 650. doi: 10.1137/120898024. Google Scholar

[16]

T. Y. Hou, X. Wu and Z. Cai, Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients,, Math. Comp., 68 (1999), 913. doi: 10.1090/S0025-5718-99-01077-7. Google Scholar

[17]

T. Y. Hou and X. H. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media,, Journal of Computational Physics, 134 (1997), 169. doi: 10.1006/jcph.1997.5682. Google Scholar

[18]

T. J. Hughes, G. R. Feijóo, L. Mazzei and J. Quincy, The variational multiscale method-a paradigm for computational mechanics,, Computer Methods in Applied Mechanics and Engineering, 166 (1998), 3. doi: 10.1016/S0045-7825(98)00079-6. Google Scholar

[19]

T. J. R. Hughes and G. Sangalli, Variational multiscale analysis: The fine-scale Green's function, projection, optimization, localization, and stabilized methods,, SIAM J. Numer. Anal., 45 (2007), 539. doi: 10.1137/050645646. Google Scholar

[20]

C. L. Bris, F. Legoll and A. Lozinski, An msfem type approach for perforated domains,, Multiscale Model. Simul., 12 (2014), 1046. doi: 10.1137/130927826. Google Scholar

[21]

C. Le Bris, F. Legoll and A. Lozinski, Msfem à la crouzeix-raviart for highly oscillatory elliptic problems,, In Philippe G. Ciarlet, (2014), 265. doi: 10.1007/978-3-642-41401-5_11. Google Scholar

[22]

J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, volume 1., Springer Science & Business Media, (2012). Google Scholar

[23]

A. Målqvist and D. Peterseim, Localization of elliptic multiscale problems,, Math. Comp., 83 (2014), 2583. doi: 10.1090/S0025-5718-2014-02868-8. Google Scholar

[24]

E. Marusic-Paloka and A. Mikelic, An error estimate for correctors in the homogenization of the Stokes and Navier-Stokes equations in a porous medium,, Bollettino U.M.I, 7 (1996), 661. Google Scholar

[25]

V. Maz'ya and T. O. Shaposhnikova, Sobolev Spaces: With Applications to Elliptic Partial Differential Equations, volume 342., Springer, (2011). doi: 10.1007/978-3-642-15564-2. Google Scholar

[26]

V. G. Maz'ya, Classes of domains and imbedding theorems for function spaces,, Soviet Math. Dokl, 1 (1960), 882. Google Scholar

[27]

S. Moskow and M. Vogelius, First-order corrections to the homogenised eigenvalues of a periodic composite medium. a convergence proof,, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 127 (1997), 1263. doi: 10.1017/S0308210500027050. Google Scholar

[28]

L. Nirenberg, On elliptic partial differential equations,, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, 13 (1959), 115. Google Scholar

[29]

C. Pechstein and R. Scheichl, Weighted poincaré inequalities,, IMA Journal of Numerical Analysis, 33 (2013), 652. doi: 10.1093/imanum/drs017. Google Scholar

[30]

B. Putra Muljadi, J. Narski, A. Lozinski and P. Degond, Non-conforming multiscale finite element method for stokes flows in heterogeneous media. part I: Methodologies and numerical experiments,, Multiscale Model. Simul., 13 (2015), 1146. doi: 10.1137/14096428X. Google Scholar

[31]

E. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory, volume 127 of Lecture Notes in Physics,, Springer-Verlag, (1980). Google Scholar

show all references

References:
[1]

A. Abdulle, W. E, B. Engquist and E. Vanden-Eijnden, The heterogeneous multiscale method,, Acta Numer., 21 (2012), 1. Google Scholar

[2]

S. C. Brenner and R. S. Scott, The Mathematical Theory of Finite Element Methods, volume 15 of Texts in Applied Mathematics,, Springer, (2008). doi: 10.1007/978-0-387-75934-0. Google Scholar

[3]

C. L. Bris, F. Legoll and A. Lozinski, An MsFEM type approach for perforated domains,, Multiscale Model. Simul., 12 (2014), 1046. doi: 10.1137/130927826. Google Scholar

[4]

D. L. Brown and D. Peterseim, A Multiscale Method for Porous Microstructures,, Multiscale Model. Simul., 14 (2016), 1123. doi: 10.1137/140995210. Google Scholar

[5]

G. A. Chechkin, A. L. Piatniski and A. S. Shamev, Homogenization: Methods and Applications, volume 234 of Translations of Mathematical Monographs,, American Mathematical Society, (2007). Google Scholar

[6]

D. Cioranescu, A. Damlamian and G. Griso, The periodic unfolding method in homogenization,, SIAM Journal on Mathematical Analysis, 40 (2008), 1585. doi: 10.1137/080713148. Google Scholar

[7]

D. Cioranescu, A. Damlamian, G. Griso and D. Onofrei, The periodic unfolding method for perforated domains and neumann sieve models,, Journal de Mathématiques Pures et Appliquées, 89 (2008), 248. doi: 10.1016/j.matpur.2007.12.008. Google Scholar

[8]

W. E and B. Engquist, The heterogeneous multiscale methods,, Commun. Math. Sci., 1 (2003), 87. Google Scholar

[9]

Y. Efendiev, T. Hou and V. Ginting, Multiscale finite element methods for nonlinear problems and their applications,, Commun. Math. Sci., 2 (2004), 553. Google Scholar

[10]

Y. R. Efendiev and X. Wu, Multiscale finite element for problems with highly oscillatory coefficients,, Numer. Math., 90 (2002), 459. doi: 10.1007/s002110100274. Google Scholar

[11]

Y. R. Efendiev, T. Y. Hou and X.-H. Wu, Convergence of a nonconforming multiscale finite element method,, SIAM Journal on Numerical Analysis, 37 (2000), 888. doi: 10.1137/S0036142997330329. Google Scholar

[12]

G. Griso, Error estimate and unfolding for periodic homogenization,, C. R. Math. Acad. Sci. Paris, 335 (2002), 333. doi: 10.1016/S1631-073X(02)02477-9. Google Scholar

[13]

P. Henning, P. Morgenstern and D. Peterseim, Multiscale Partition of Unity,, In M. Griebel and M. A. Schweitzer, (2014). Google Scholar

[14]

P. Henning and M. Ohlberger, The heterogeneous multiscale finite element method for elliptic homogenization problems in perforated domains,, Numer. Math, 113 (2009), 601. doi: 10.1007/s00211-009-0244-4. Google Scholar

[15]

J. S. Hesthaven, S. Zhang and X. Zhu, High-order multiscale finite element method for elliptic problems,, Multiscale Modeling & Simulation, 12 (2014), 650. doi: 10.1137/120898024. Google Scholar

[16]

T. Y. Hou, X. Wu and Z. Cai, Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients,, Math. Comp., 68 (1999), 913. doi: 10.1090/S0025-5718-99-01077-7. Google Scholar

[17]

T. Y. Hou and X. H. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media,, Journal of Computational Physics, 134 (1997), 169. doi: 10.1006/jcph.1997.5682. Google Scholar

[18]

T. J. Hughes, G. R. Feijóo, L. Mazzei and J. Quincy, The variational multiscale method-a paradigm for computational mechanics,, Computer Methods in Applied Mechanics and Engineering, 166 (1998), 3. doi: 10.1016/S0045-7825(98)00079-6. Google Scholar

[19]

T. J. R. Hughes and G. Sangalli, Variational multiscale analysis: The fine-scale Green's function, projection, optimization, localization, and stabilized methods,, SIAM J. Numer. Anal., 45 (2007), 539. doi: 10.1137/050645646. Google Scholar

[20]

C. L. Bris, F. Legoll and A. Lozinski, An msfem type approach for perforated domains,, Multiscale Model. Simul., 12 (2014), 1046. doi: 10.1137/130927826. Google Scholar

[21]

C. Le Bris, F. Legoll and A. Lozinski, Msfem à la crouzeix-raviart for highly oscillatory elliptic problems,, In Philippe G. Ciarlet, (2014), 265. doi: 10.1007/978-3-642-41401-5_11. Google Scholar

[22]

J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, volume 1., Springer Science & Business Media, (2012). Google Scholar

[23]

A. Målqvist and D. Peterseim, Localization of elliptic multiscale problems,, Math. Comp., 83 (2014), 2583. doi: 10.1090/S0025-5718-2014-02868-8. Google Scholar

[24]

E. Marusic-Paloka and A. Mikelic, An error estimate for correctors in the homogenization of the Stokes and Navier-Stokes equations in a porous medium,, Bollettino U.M.I, 7 (1996), 661. Google Scholar

[25]

V. Maz'ya and T. O. Shaposhnikova, Sobolev Spaces: With Applications to Elliptic Partial Differential Equations, volume 342., Springer, (2011). doi: 10.1007/978-3-642-15564-2. Google Scholar

[26]

V. G. Maz'ya, Classes of domains and imbedding theorems for function spaces,, Soviet Math. Dokl, 1 (1960), 882. Google Scholar

[27]

S. Moskow and M. Vogelius, First-order corrections to the homogenised eigenvalues of a periodic composite medium. a convergence proof,, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 127 (1997), 1263. doi: 10.1017/S0308210500027050. Google Scholar

[28]

L. Nirenberg, On elliptic partial differential equations,, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, 13 (1959), 115. Google Scholar

[29]

C. Pechstein and R. Scheichl, Weighted poincaré inequalities,, IMA Journal of Numerical Analysis, 33 (2013), 652. doi: 10.1093/imanum/drs017. Google Scholar

[30]

B. Putra Muljadi, J. Narski, A. Lozinski and P. Degond, Non-conforming multiscale finite element method for stokes flows in heterogeneous media. part I: Methodologies and numerical experiments,, Multiscale Model. Simul., 13 (2015), 1146. doi: 10.1137/14096428X. Google Scholar

[31]

E. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory, volume 127 of Lecture Notes in Physics,, Springer-Verlag, (1980). Google Scholar

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