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.

[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.

[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.

[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.

[5]

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

[6]

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

[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.

[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.

[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.

[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.

[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.

[12]

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

[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.

[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.

[15]

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

[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.

[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.

[18]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[26]

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

[27]

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

[28]

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

[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.

[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.

[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.

[32]

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

[33]

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

[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.

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.

[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.

[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.

[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.

[5]

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

[6]

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

[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.

[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.

[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.

[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.

[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.

[12]

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

[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.

[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.

[15]

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

[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.

[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.

[18]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[26]

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

[27]

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

[28]

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

[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.

[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.

[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.

[32]

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

[33]

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

[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.

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