October  2016, 9(5): 1269-1298. doi: 10.3934/dcdss.2016051

Multiscale mixed finite elements

1. 

Department of Information Technology, Uppsala University, Box 337, SE-751 05 Uppsala, Sweden

2. 

Department of Mathematics, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden

3. 

Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, SE-412 96 Göteborg, Sweden

Received  January 2015 Revised  August 2015 Published  October 2016

In this work, we propose a mixed finite element method for solving elliptic multiscale problems based on a localized orthogonal decomposition (LOD) of Raviart--Thomas finite element spaces. It requires to solve local problems in small patches around the elements of a coarse grid. These computations can be perfectly parallelized and are cheap to perform. Using the results of these patch problems, we construct a low dimensional multiscale mixed finite element space with very high approximation properties. This space can be used for solving the original saddle point problem in an efficient way. We prove convergence of our approach, independent of structural assumptions or scale separation. Finally, we demonstrate the applicability of our method by presenting a variety of numerical experiments, including a comparison with an MsFEM approach.
Citation: Fredrik Hellman, Patrick Henning, Axel Målqvist. Multiscale mixed finite elements. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1269-1298. doi: 10.3934/dcdss.2016051
References:
[1]

J. Aarnes, On the use of a mixed multiscale finite element method for greater flexibility and increased speed or improved accuracy in reservoir simulation,, Multiscale Model. Simul., 2 (2004), 421. doi: 10.1137/030600655. Google Scholar

[2]

A. Abdulle and P. Henning, A reduced basis localized orthogonal decomposition,, J. Comput. Phys., 295 (2015), 379. doi: 10.1016/j.jcp.2015.04.016. Google Scholar

[3]

A. Abdulle and P. Henning, Localized orthogonal decomposition method for the wave equation with a continuum of scales,, to appear in Math. Comp., (2016). doi: 10.1090/mcom/3114. Google Scholar

[4]

T. Arbogast, Analysis of a two-scale, locally conservative subgrid upscaling for elliptic problems,, SIAM J. Numer. Anal., 42 (2004), 576. doi: 10.1137/S0036142902406636. Google Scholar

[5]

T. Arbogast, Homogenization-based mixed multiscale finite elements for problems with anisotropy,, Multiscale Model. Simul., 9 (2011), 624. doi: 10.1137/100788677. Google Scholar

[6]

T. Arbogast and K. Boyd, Subgrid upscaling and mixed multiscale finite elements,, SIAM J. Numer. Anal., 44 (2006), 1150. doi: 10.1137/050631811. Google Scholar

[7]

D. N. Arnold, R. S. Falk and R. Winther, Finite element exterior calculus, homological techniques, and applications,, Acta Numer., 15 (2006), 1. doi: 10.1017/S0962492906210018. Google Scholar

[8]

D. Boffi, F. Brezzi and M. Fortin, Mixed Finite Element Methods and Applications, volume 44 of {Springer Series in Computational Mathematics,, Springer-Verlag, (2013). doi: 10.1007/978-3-642-36519-5. Google Scholar

[9]

Z. Chen and T. Y. Hou, A mixed multiscale finite element method for elliptic problems with oscillating coefficients,, Math. Comp., 72 (2003), 541. doi: 10.1090/S0025-5718-02-01441-2. Google Scholar

[10]

S. H. Christiansen, Stability of Hodge decompositions in finite element spaces of differential forms in arbitrary dimension,, Numer. Math., 107 (2007), 87. doi: 10.1007/s00211-007-0081-2. Google Scholar

[11]

S. H. Christiansen and R. Winther, Smoothed projections in finite element exterior calculus,, Math. Comp., 77 (2008), 813. doi: 10.1090/S0025-5718-07-02081-9. Google Scholar

[12]

M. A. Christie, Tenth SPE comparative solution project: A comparison of upscaling techniques,, SPE Reservoir Eval. Eng., 4 (2001), 308. Google Scholar

[13]

D. Elfverson, E. H. Georgoulis, A. Målqvist and D. Peterseim, Convergence of a discontinuous Galerkin multiscale method,, SIAM J. Numer. Anal., 51 (2013), 3351. doi: 10.1137/120900113. Google Scholar

[14]

D. Elfverson, V. Ginting and P. Henning, On multiscale methods in Petrov-Galerkin formulation,, Numer. Math., 131 (2015), 643. doi: 10.1007/s00211-015-0703-z. Google Scholar

[15]

P. Henning and A. Målqvist, Localized orthogonal decomposition techniques for boundary value problems,, SIAM J. Sci. Comput., 36 (2014). doi: 10.1137/130933198. Google Scholar

[16]

P. Henning, A. Målqvist and D. Peterseim, A localized orthogonal decomposition method for semi-linear elliptic problems,, ESAIM Math. Model. Numer. Anal., 48 (2014), 1331. doi: 10.1051/m2an/2013141. Google Scholar

[17]

P. Henning, P. Morgenstern and D. Peterseim, Multiscale partition of unity,, In M. Griebel and M. A. Schweitzer, (2015), 185. doi: 10.1007/978-3-319-06898-5_10. Google Scholar

[18]

P. Henning and D. Peterseim, Oversampling for the multiscale finite element method,, Multiscale Model. Simul., 11 (2013), 1149. doi: 10.1137/120900332. Google Scholar

[19]

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

[20]

T. 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

[21]

T. J. R. Hughes, Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods,, Comput. Methods Appl. Mech. Engrg., 127 (1995), 387. doi: 10.1016/0045-7825(95)00844-9. Google Scholar

[22]

T. J. R. Hughes, G. R. Feijóo, L. Mazzei and J.-B. Quincy, The variational multiscale method-a paradigm for computational mechanics,, Comput. Methods Appl. Mech. Engrg., 166 (1998), 3. doi: 10.1016/S0045-7825(98)00079-6. Google Scholar

[23]

D. Iftimie, G. Karch and C. Lacave, Asymptotics of solutions to the Navier-Stokes system in exterior domains,, J. Lond. Math. Soc. (2), 90 (2014), 785. doi: 10.1112/jlms/jdu052. Google Scholar

[24]

M. G. Larson and A. Målqvist, Adaptive variational multiscale methods based on a posteriori error estimation: Energy norm estimates for elliptic problems,, Comput. Methods Appl. Mech. Engrg., 196 (2007), 2313. doi: 10.1016/j.cma.2006.08.019. Google Scholar

[25]

M. G. Larson and A. Målqvist, A mixed adaptive variational multiscale method with applications in oil reservoir simulation,, Math. Models Methods Appl. Sci., 19 (2009), 1017. doi: 10.1142/S021820250900370X. Google Scholar

[26]

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

[27]

A. Målqvist and D. Peterseim, Computation of eigenvalues by numerical upscaling,, Numer. Math., 130 (2015), 337. doi: 10.1007/s00211-014-0665-6. Google Scholar

[28]

A. Målqvist, Multiscale methods for elliptic problems,, Multiscale Model. Simul., 9 (2011), 1064. doi: 10.1137/090775592. Google Scholar

[29]

J. Nolen, G. Papanicolaou and O. Pironneau, A framework for adaptive multiscale methods for elliptic problems,, Multiscale Model. Simul., 7 (2008), 171. doi: 10.1137/070693230. Google Scholar

[30]

D. Peterseim, Eliminating the pollution effect in Helmholtz problems by local subscale correction,, to appear in Math. Comp., (2016). doi: 10.1090/mcom/3156. Google Scholar

[31]

P. A. Raviart and J. M. Thomas, A mixed finite element method for 2-nd order elliptic problems,, In I. Galligani and E. Magenes, (1977), 292. Google Scholar

[32]

J. Schöberl, A posteriori error estimates for Maxwell equations,, Math. Comp., 77 (2008), 633. doi: 10.1090/S0025-5718-07-02030-3. Google Scholar

[33]

H. Wendland, Divergence-free kernel methods for approximating the Stokes problem,, SIAM J. Numer. Anal., 47 (2009), 3158. doi: 10.1137/080730299. Google Scholar

[34]

B. Wohlmuth, A. Toselli and O. Widlund, An iterative substructuring method for Raviart-Thomas vector fields in three dimensions,, SIAM J. Numer. Anal., 37 (2000), 1657. doi: 10.1137/S0036142998347310. Google Scholar

show all references

References:
[1]

J. Aarnes, On the use of a mixed multiscale finite element method for greater flexibility and increased speed or improved accuracy in reservoir simulation,, Multiscale Model. Simul., 2 (2004), 421. doi: 10.1137/030600655. Google Scholar

[2]

A. Abdulle and P. Henning, A reduced basis localized orthogonal decomposition,, J. Comput. Phys., 295 (2015), 379. doi: 10.1016/j.jcp.2015.04.016. Google Scholar

[3]

A. Abdulle and P. Henning, Localized orthogonal decomposition method for the wave equation with a continuum of scales,, to appear in Math. Comp., (2016). doi: 10.1090/mcom/3114. Google Scholar

[4]

T. Arbogast, Analysis of a two-scale, locally conservative subgrid upscaling for elliptic problems,, SIAM J. Numer. Anal., 42 (2004), 576. doi: 10.1137/S0036142902406636. Google Scholar

[5]

T. Arbogast, Homogenization-based mixed multiscale finite elements for problems with anisotropy,, Multiscale Model. Simul., 9 (2011), 624. doi: 10.1137/100788677. Google Scholar

[6]

T. Arbogast and K. Boyd, Subgrid upscaling and mixed multiscale finite elements,, SIAM J. Numer. Anal., 44 (2006), 1150. doi: 10.1137/050631811. Google Scholar

[7]

D. N. Arnold, R. S. Falk and R. Winther, Finite element exterior calculus, homological techniques, and applications,, Acta Numer., 15 (2006), 1. doi: 10.1017/S0962492906210018. Google Scholar

[8]

D. Boffi, F. Brezzi and M. Fortin, Mixed Finite Element Methods and Applications, volume 44 of {Springer Series in Computational Mathematics,, Springer-Verlag, (2013). doi: 10.1007/978-3-642-36519-5. Google Scholar

[9]

Z. Chen and T. Y. Hou, A mixed multiscale finite element method for elliptic problems with oscillating coefficients,, Math. Comp., 72 (2003), 541. doi: 10.1090/S0025-5718-02-01441-2. Google Scholar

[10]

S. H. Christiansen, Stability of Hodge decompositions in finite element spaces of differential forms in arbitrary dimension,, Numer. Math., 107 (2007), 87. doi: 10.1007/s00211-007-0081-2. Google Scholar

[11]

S. H. Christiansen and R. Winther, Smoothed projections in finite element exterior calculus,, Math. Comp., 77 (2008), 813. doi: 10.1090/S0025-5718-07-02081-9. Google Scholar

[12]

M. A. Christie, Tenth SPE comparative solution project: A comparison of upscaling techniques,, SPE Reservoir Eval. Eng., 4 (2001), 308. Google Scholar

[13]

D. Elfverson, E. H. Georgoulis, A. Målqvist and D. Peterseim, Convergence of a discontinuous Galerkin multiscale method,, SIAM J. Numer. Anal., 51 (2013), 3351. doi: 10.1137/120900113. Google Scholar

[14]

D. Elfverson, V. Ginting and P. Henning, On multiscale methods in Petrov-Galerkin formulation,, Numer. Math., 131 (2015), 643. doi: 10.1007/s00211-015-0703-z. Google Scholar

[15]

P. Henning and A. Målqvist, Localized orthogonal decomposition techniques for boundary value problems,, SIAM J. Sci. Comput., 36 (2014). doi: 10.1137/130933198. Google Scholar

[16]

P. Henning, A. Målqvist and D. Peterseim, A localized orthogonal decomposition method for semi-linear elliptic problems,, ESAIM Math. Model. Numer. Anal., 48 (2014), 1331. doi: 10.1051/m2an/2013141. Google Scholar

[17]

P. Henning, P. Morgenstern and D. Peterseim, Multiscale partition of unity,, In M. Griebel and M. A. Schweitzer, (2015), 185. doi: 10.1007/978-3-319-06898-5_10. Google Scholar

[18]

P. Henning and D. Peterseim, Oversampling for the multiscale finite element method,, Multiscale Model. Simul., 11 (2013), 1149. doi: 10.1137/120900332. Google Scholar

[19]

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

[20]

T. 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

[21]

T. J. R. Hughes, Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods,, Comput. Methods Appl. Mech. Engrg., 127 (1995), 387. doi: 10.1016/0045-7825(95)00844-9. Google Scholar

[22]

T. J. R. Hughes, G. R. Feijóo, L. Mazzei and J.-B. Quincy, The variational multiscale method-a paradigm for computational mechanics,, Comput. Methods Appl. Mech. Engrg., 166 (1998), 3. doi: 10.1016/S0045-7825(98)00079-6. Google Scholar

[23]

D. Iftimie, G. Karch and C. Lacave, Asymptotics of solutions to the Navier-Stokes system in exterior domains,, J. Lond. Math. Soc. (2), 90 (2014), 785. doi: 10.1112/jlms/jdu052. Google Scholar

[24]

M. G. Larson and A. Målqvist, Adaptive variational multiscale methods based on a posteriori error estimation: Energy norm estimates for elliptic problems,, Comput. Methods Appl. Mech. Engrg., 196 (2007), 2313. doi: 10.1016/j.cma.2006.08.019. Google Scholar

[25]

M. G. Larson and A. Målqvist, A mixed adaptive variational multiscale method with applications in oil reservoir simulation,, Math. Models Methods Appl. Sci., 19 (2009), 1017. doi: 10.1142/S021820250900370X. Google Scholar

[26]

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

[27]

A. Målqvist and D. Peterseim, Computation of eigenvalues by numerical upscaling,, Numer. Math., 130 (2015), 337. doi: 10.1007/s00211-014-0665-6. Google Scholar

[28]

A. Målqvist, Multiscale methods for elliptic problems,, Multiscale Model. Simul., 9 (2011), 1064. doi: 10.1137/090775592. Google Scholar

[29]

J. Nolen, G. Papanicolaou and O. Pironneau, A framework for adaptive multiscale methods for elliptic problems,, Multiscale Model. Simul., 7 (2008), 171. doi: 10.1137/070693230. Google Scholar

[30]

D. Peterseim, Eliminating the pollution effect in Helmholtz problems by local subscale correction,, to appear in Math. Comp., (2016). doi: 10.1090/mcom/3156. Google Scholar

[31]

P. A. Raviart and J. M. Thomas, A mixed finite element method for 2-nd order elliptic problems,, In I. Galligani and E. Magenes, (1977), 292. Google Scholar

[32]

J. Schöberl, A posteriori error estimates for Maxwell equations,, Math. Comp., 77 (2008), 633. doi: 10.1090/S0025-5718-07-02030-3. Google Scholar

[33]

H. Wendland, Divergence-free kernel methods for approximating the Stokes problem,, SIAM J. Numer. Anal., 47 (2009), 3158. doi: 10.1137/080730299. Google Scholar

[34]

B. Wohlmuth, A. Toselli and O. Widlund, An iterative substructuring method for Raviart-Thomas vector fields in three dimensions,, SIAM J. Numer. Anal., 37 (2000), 1657. doi: 10.1137/S0036142998347310. Google Scholar

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