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

[1]

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

[2]

Ferenc Weisz. Dual spaces of mixed-norm martingale hardy spaces. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020285

[3]

Ying Liu, Yanping Chen, Yunqing Huang, Yang Wang. Two-grid method for semiconductor device problem by mixed finite element method and characteristics finite element method. Electronic Research Archive, 2021, 29 (1) : 1859-1880. doi: 10.3934/era.2020095

[4]

Wenya Qi, Padmanabhan Seshaiyer, Junping Wang. A four-field mixed finite element method for Biot's consolidation problems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020127

[5]

Alain Damlamian, Klas Pettersson. Homogenization of oscillating boundaries. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 197-219. doi: 10.3934/dcds.2009.23.197

[6]

Thomas Y. Hou, Dong Liang. Multiscale analysis for convection dominated transport equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 281-298. doi: 10.3934/dcds.2009.23.281

[7]

Monia Capanna, Jean C. Nakasato, Marcone C. Pereira, Julio D. Rossi. Homogenization for nonlocal problems with smooth kernels. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020385

[8]

Eduard Marušić-Paloka, Igor Pažanin. Homogenization and singular perturbation in porous media. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020279

[9]

Martin Heida, Stefan Neukamm, Mario Varga. Stochastic homogenization of $ \Lambda $-convex gradient flows. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 427-453. doi: 10.3934/dcdss.2020328

[10]

John Mallet-Paret, Roger D. Nussbaum. Asymptotic homogenization for delay-differential equations and a question of analyticity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3789-3812. doi: 10.3934/dcds.2020044

[11]

George W. Patrick. The geometry of convergence in numerical analysis. Journal of Computational Dynamics, 2021, 8 (1) : 33-58. doi: 10.3934/jcd.2021003

[12]

Alessandro Carbotti, Giovanni E. Comi. A note on Riemann-Liouville fractional Sobolev spaces. Communications on Pure & Applied Analysis, 2021, 20 (1) : 17-54. doi: 10.3934/cpaa.2020255

[13]

Giulia Cavagnari, Antonio Marigonda. Attainability property for a probabilistic target in wasserstein spaces. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 777-812. doi: 10.3934/dcds.2020300

[14]

Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249

[15]

Jie Zhang, Yuping Duan, Yue Lu, Michael K. Ng, Huibin Chang. Bilinear constraint based ADMM for mixed Poisson-Gaussian noise removal. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020071

[16]

Jianli Xiang, Guozheng Yan. The uniqueness of the inverse elastic wave scattering problem based on the mixed reciprocity relation. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021004

[17]

Riadh Chteoui, Abdulrahman F. Aljohani, Anouar Ben Mabrouk. Classification and simulation of chaotic behaviour of the solutions of a mixed nonlinear Schrödinger system. Electronic Research Archive, , () : -. doi: 10.3934/era.2021002

[18]

Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, 2021, 14 (1) : 115-148. doi: 10.3934/krm.2020051

[19]

Federico Rodriguez Hertz, Zhiren Wang. On $ \epsilon $-escaping trajectories in homogeneous spaces. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 329-357. doi: 10.3934/dcds.2020365

[20]

Noah Stevenson, Ian Tice. A truncated real interpolation method and characterizations of screened Sobolev spaces. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5509-5566. doi: 10.3934/cpaa.2020250

2019 Impact Factor: 1.233

Metrics

  • PDF downloads (86)
  • HTML views (0)
  • Cited by (5)

[Back to Top]