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 and 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), 421-439. doi: 10.1137/030600655.

[2]

A. Abdulle and P. Henning, A reduced basis localized orthogonal decomposition, J. Comput. Phys., 295 (2015), 379-401. 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, arXiv:1406.6325. 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), 576-598 (electronic). doi: 10.1137/S0036142902406636.

[5]

T. Arbogast, Homogenization-based mixed multiscale finite elements for problems with anisotropy, Multiscale Model. Simul., 9 (2011), 624-653. doi: 10.1137/100788677.

[6]

T. Arbogast and K. Boyd, Subgrid upscaling and mixed multiscale finite elements, SIAM J. Numer. Anal., 44 (2006), 1150-1171. 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), 1-155. 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, Berlin Heidelberg, 2nd edition, 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), 541-576. 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), 87-106. 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), 813-829. 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), 308-317.

[13]

D. Elfverson, E. H. Georgoulis, A. Målqvist and D. Peterseim, Convergence of a discontinuous Galerkin multiscale method, SIAM J. Numer. Anal., 51 (2013), 3351-3372. doi: 10.1137/120900113.

[14]

D. Elfverson, V. Ginting and P. Henning, On multiscale methods in Petrov-Galerkin formulation, Numer. Math., 131 (2015), 643-682. 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), A1609-A1634. 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), 1331-1349. doi: 10.1051/m2an/2013141.

[17]

P. Henning, P. Morgenstern and D. Peterseim, Multiscale partition of unity, In M. Griebel and M. A. Schweitzer, editors, Meshfree Methods for Partial Differential Equations VII, volume 100 of Lecture Notes in Computational Science and Engineering, pages 185-204. Springer International Publishing, 2015. 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), 1149-1175. 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), 169-189. 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), 539-557. 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), 387-401. 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), 3-24. 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), 785-806. 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), 2313-2324. 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), 1017-1042. doi: 10.1142/S021820250900370X.

[26]

A. Målqvist and D. Peterseim, Localization of elliptic multiscale problems, Math. Comp., 83 (2014), 2583-2603. doi: 10.1090/S0025-5718-2014-02868-8.

[27]

A. Målqvist and D. Peterseim, Computation of eigenvalues by numerical upscaling, Numer. Math., 130 (2015), 337-361. doi: 10.1007/s00211-014-0665-6.

[28]

A. Målqvist, Multiscale methods for elliptic problems, Multiscale Model. Simul., 9 (2011), 1064-1086. 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), 171-196. 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, editors, Mathematical Aspects of Finite Element Methods, volume 606 of Lecture Notes in Mathematics, pages 292-315. Springer Berlin Heidelberg, 1977.

[32]

J. Schöberl, A posteriori error estimates for Maxwell equations, Math. Comp., 77 (2008), 633-649. 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), 3158-3179. 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), 1657-1676. 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), 421-439. doi: 10.1137/030600655.

[2]

A. Abdulle and P. Henning, A reduced basis localized orthogonal decomposition, J. Comput. Phys., 295 (2015), 379-401. 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, arXiv:1406.6325. 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), 576-598 (electronic). doi: 10.1137/S0036142902406636.

[5]

T. Arbogast, Homogenization-based mixed multiscale finite elements for problems with anisotropy, Multiscale Model. Simul., 9 (2011), 624-653. doi: 10.1137/100788677.

[6]

T. Arbogast and K. Boyd, Subgrid upscaling and mixed multiscale finite elements, SIAM J. Numer. Anal., 44 (2006), 1150-1171. 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), 1-155. 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, Berlin Heidelberg, 2nd edition, 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), 541-576. 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), 87-106. 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), 813-829. 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), 308-317.

[13]

D. Elfverson, E. H. Georgoulis, A. Målqvist and D. Peterseim, Convergence of a discontinuous Galerkin multiscale method, SIAM J. Numer. Anal., 51 (2013), 3351-3372. doi: 10.1137/120900113.

[14]

D. Elfverson, V. Ginting and P. Henning, On multiscale methods in Petrov-Galerkin formulation, Numer. Math., 131 (2015), 643-682. 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), A1609-A1634. 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), 1331-1349. doi: 10.1051/m2an/2013141.

[17]

P. Henning, P. Morgenstern and D. Peterseim, Multiscale partition of unity, In M. Griebel and M. A. Schweitzer, editors, Meshfree Methods for Partial Differential Equations VII, volume 100 of Lecture Notes in Computational Science and Engineering, pages 185-204. Springer International Publishing, 2015. 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), 1149-1175. 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), 169-189. 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), 539-557. 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), 387-401. 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), 3-24. 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), 785-806. 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), 2313-2324. 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), 1017-1042. doi: 10.1142/S021820250900370X.

[26]

A. Målqvist and D. Peterseim, Localization of elliptic multiscale problems, Math. Comp., 83 (2014), 2583-2603. doi: 10.1090/S0025-5718-2014-02868-8.

[27]

A. Målqvist and D. Peterseim, Computation of eigenvalues by numerical upscaling, Numer. Math., 130 (2015), 337-361. doi: 10.1007/s00211-014-0665-6.

[28]

A. Målqvist, Multiscale methods for elliptic problems, Multiscale Model. Simul., 9 (2011), 1064-1086. 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), 171-196. 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, editors, Mathematical Aspects of Finite Element Methods, volume 606 of Lecture Notes in Mathematics, pages 292-315. Springer Berlin Heidelberg, 1977.

[32]

J. Schöberl, A posteriori error estimates for Maxwell equations, Math. Comp., 77 (2008), 633-649. 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), 3158-3179. 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), 1657-1676. doi: 10.1137/S0036142998347310.

[1]

Tianliang Hou, Yanping Chen. Superconvergence for elliptic optimal control problems discretized by RT1 mixed finite elements and linear discontinuous elements. Journal of Industrial and Management Optimization, 2013, 9 (3) : 631-642. doi: 10.3934/jimo.2013.9.631

[2]

Antoine Gloria Cermics. A direct approach to numerical homogenization in finite elasticity. Networks and Heterogeneous Media, 2006, 1 (1) : 109-141. doi: 10.3934/nhm.2006.1.109

[3]

Dag Lukkassen, Annette Meidell, Peter Wall. Multiscale homogenization of monotone operators. Discrete and Continuous Dynamical Systems, 2008, 22 (3) : 711-727. doi: 10.3934/dcds.2008.22.711

[4]

Assyr Abdulle, Yun Bai, Gilles Vilmart. Reduced basis finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems. Discrete and Continuous Dynamical Systems - S, 2015, 8 (1) : 91-118. doi: 10.3934/dcdss.2015.8.91

[5]

Nils Svanstedt. Multiscale stochastic homogenization of monotone operators. Networks and Heterogeneous Media, 2007, 2 (1) : 181-192. doi: 10.3934/nhm.2007.2.181

[6]

Peter Monk, Jiguang Sun. Inverse scattering using finite elements and gap reciprocity. Inverse Problems and Imaging, 2007, 1 (4) : 643-660. doi: 10.3934/ipi.2007.1.643

[7]

Eric Dubach, Robert Luce, Jean-Marie Thomas. Pseudo-Conform Polynomial Lagrange Finite Elements on Quadrilaterals and Hexahedra. Communications on Pure and Applied Analysis, 2009, 8 (1) : 237-254. doi: 10.3934/cpaa.2009.8.237

[8]

Murat Uzunca, Ayşe Sarıaydın-Filibelioǧlu. Adaptive discontinuous galerkin finite elements for advective Allen-Cahn equation. Numerical Algebra, Control and Optimization, 2021, 11 (2) : 269-281. doi: 10.3934/naco.2020025

[9]

Zhangxin Chen, Qiaoyuan Jiang, Yanli Cui. Locking-free nonconforming finite elements for planar linear elasticity. Conference Publications, 2005, 2005 (Special) : 181-189. doi: 10.3934/proc.2005.2005.181

[10]

Emmanuel Frénod. Homogenization-based numerical methods. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : i-ix. doi: 10.3934/dcdss.201605i

[11]

Thierry Colin, Boniface Nkonga. Multiscale numerical method for nonlinear Maxwell equations. Discrete and Continuous Dynamical Systems - B, 2005, 5 (3) : 631-658. doi: 10.3934/dcdsb.2005.5.631

[12]

Fabio Camilli, Claudio Marchi. On the convergence rate in multiscale homogenization of fully nonlinear elliptic problems. Networks and Heterogeneous Media, 2011, 6 (1) : 61-75. doi: 10.3934/nhm.2011.6.61

[13]

Joel Fotso Tachago, Giuliano Gargiulo, Hubert Nnang, Elvira Zappale. Multiscale homogenization of integral convex functionals in Orlicz Sobolev setting. Evolution Equations and Control Theory, 2021, 10 (2) : 297-320. doi: 10.3934/eect.2020067

[14]

Emmanuel Frénod. An attempt at classifying homogenization-based numerical methods. Discrete and Continuous Dynamical Systems - S, 2015, 8 (1) : i-vi. doi: 10.3934/dcdss.2015.8.1i

[15]

Olivier Pironneau, Alexei Lozinski, Alain Perronnet, Frédéric Hecht. Numerical zoom for multiscale problems with an application to flows through porous media. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 265-280. doi: 10.3934/dcds.2009.23.265

[16]

Annalisa Iuorio, Christian Kuehn, Peter Szmolyan. Geometry and numerical continuation of multiscale orbits in a nonconvex variational problem. Discrete and Continuous Dynamical Systems - S, 2020, 13 (4) : 1269-1290. doi: 10.3934/dcdss.2020073

[17]

Alberto Zingaro, Ivan Fumagalli, Luca Dede, Marco Fedele, Pasquale C. Africa, Antonio F. Corno, Alfio Quarteroni. A geometric multiscale model for the numerical simulation of blood flow in the human left heart. Discrete and Continuous Dynamical Systems - S, 2022, 15 (8) : 2391-2427. doi: 10.3934/dcdss.2022052

[18]

X.H. Wu, Y. Efendiev, Thomas Y. Hou. Analysis of upscaling absolute permeability. Discrete and Continuous Dynamical Systems - B, 2002, 2 (2) : 185-204. doi: 10.3934/dcdsb.2002.2.185

[19]

Donald L. Brown, Vasilena Taralova. A multiscale finite element method for Neumann problems in porous microstructures. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : 1299-1326. doi: 10.3934/dcdss.2016052

[20]

Marcin Studniarski. Finding all minimal elements of a finite partially ordered set by genetic algorithm with a prescribed probability. Numerical Algebra, Control and Optimization, 2011, 1 (3) : 389-398. doi: 10.3934/naco.2011.1.389

2020 Impact Factor: 2.425

Metrics

  • PDF downloads (278)
  • HTML views (0)
  • Cited by (6)

[Back to Top]