Non-intrusive implementation of a wide variety of Multiscale Finite Element Methods

Rutger A. Biezemans; Claude Le Bris; Frédéric Legoll; Alexei Lozinski

doi:10.5802/crmeca.178

Non-intrusive implementation of a wide variety of Multiscale Finite Element Methods

Rutger A. Biezemans ^1,² ; Claude Le Bris ^1,² ; Frédéric Legoll ^1,² ; Alexei Lozinski ^2,³

¹ École Nationale des Ponts et Chaussées, 6 et 8 avenue Blaise Pascal, 77455 Marne-La-Vallée Cedex 2, France
² MATHERIALS project-team, Inria Paris, 2 rue Simone Iff, CS 42112, 75589 Paris Cedex 12, France
³ Université de Franche-Comté, CNRS, LmB, F-25000 Besançon, France

Comptes Rendus. Mécanique, Volume 351 (2023) no. S1, pp. 135-180.

Résumé

Multiscale Finite Element Methods (MsFEMs) are now well-established finite element type approaches dedicated to multiscale problems. They first compute local, oscillatory, problem-dependent basis functions that generate a suitable discretization space, and next perform a Galerkin approximation of the problem on that space. We investigate here how these approaches can be implemented in a non-intrusive way, in order to facilitate their dissemination within industrial codes or non-academic environments. We develop an abstract framework that covers a wide variety of MsFEMs for linear second-order partial differential equations. Non-intrusive MsFEM approaches are developed within the full generality of this framework, which may moreover be beneficial to steering software development and improving the theoretical understanding and analysis of MsFEMs.

Reçu le : 2022-11-30
Révisé le : 2023-02-01
Accepté le : 2023-02-06
Première publication : 2023-07-27
Publié le : 2024-04-26

DOI : 10.5802/crmeca.178

Mots clés : Partial differential equations, Finite element methods, Multiscale problems, Non-intrusive implementation

Affiliations des auteurs :

Rutger A. Biezemans ^{1, 2} ; Claude Le Bris ^{1, 2} ; Frédéric Legoll ^{1, 2} ; Alexei Lozinski ^{2, 3}

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMECA_2023__351_S1_135_0,
     author = {Rutger A. Biezemans and Claude Le Bris and Fr\'ed\'eric Legoll and Alexei Lozinski},
     title = {Non-intrusive implementation of a wide variety of {Multiscale} {Finite} {Element} {Methods}},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {135--180},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {351},
     number = {S1},
     year = {2023},
     doi = {10.5802/crmeca.178},
     language = {en},
}

TY  - JOUR
AU  - Rutger A. Biezemans
AU  - Claude Le Bris
AU  - Frédéric Legoll
AU  - Alexei Lozinski
TI  - Non-intrusive implementation of a wide variety of Multiscale Finite Element Methods
JO  - Comptes Rendus. Mécanique
PY  - 2023
SP  - 135
EP  - 180
VL  - 351
IS  - S1
PB  - Académie des sciences, Paris
DO  - 10.5802/crmeca.178
LA  - en
ID  - CRMECA_2023__351_S1_135_0
ER  -

%0 Journal Article
%A Rutger A. Biezemans
%A Claude Le Bris
%A Frédéric Legoll
%A Alexei Lozinski
%T Non-intrusive implementation of a wide variety of Multiscale Finite Element Methods
%J Comptes Rendus. Mécanique
%D 2023
%P 135-180
%V 351
%N S1
%I Académie des sciences, Paris
%R 10.5802/crmeca.178
%G en
%F CRMECA_2023__351_S1_135_0

Rutger A. Biezemans; Claude Le Bris; Frédéric Legoll; Alexei Lozinski. Non-intrusive implementation of a wide variety of Multiscale Finite Element Methods. Comptes Rendus. Mécanique, Volume 351 (2023) no. S1, pp. 135-180. doi : 10.5802/crmeca.178. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.178/

Bibliographie
Cité par

[1] Robert Altmann; Patrick Henning; Daniel Peterseim Numerical homogenization beyond scale separation, Acta Numer., Volume 30 (2021), pp. 1-86 | DOI | MR | Zbl

[2] Weinan E; Björn Engquist The Heterogeneous Multiscale Methods, Commun. Math. Sci., Volume 1 (2003) no. 1, pp. 87-132 | DOI | MR | Zbl

[3] Axel Målqvist; Daniel Peterseim Localization of elliptic multiscale problems, Math. Comput., Volume 83 (2014) no. 290, pp. 2583-2603 | DOI | MR | Zbl

[4] Thomas Y. Hou; Xiao-Hui Wu A Multiscale Finite Element Method for Elliptic Problems in Composite Materials and Porous Media, J. Comput. Phys., Volume 134 (1997) no. 1, pp. 169-189 | DOI | MR | Zbl

[5] Assyr Abdulle; E Weinan; Björn Engquist; Eric Vanden-Eijnden The heterogeneous multiscale method, Acta Numer., Volume 21 (2012), pp. 1-87 | DOI | MR | Zbl

[6] Yalchin Efendiev; Thomas Y. Hou Multiscale Finite Element Methods, Surveys and Tutorials in the Applied Mathematical Sciences, 4, Springer, 2009 | DOI | Zbl

[7] Claude Le Bris; Frédéric Legoll Examples of computational approaches for elliptic, possibly multiscale PDEs with random inputs, J. Comput. Phys., Volume 328 (2017), pp. 455-473 | DOI | MR | Zbl

[8] Ivo Babuška; John E. Osborn Generalized Finite Element Methods: Their Performance and Their Relation to Mixed Methods, SIAM J. Numer. Anal., Volume 20 (1983) no. 3, pp. 510-536 | DOI | MR | Zbl

[9] Claude Le Bris; Frédéric Legoll; Alexei Lozinski MsFEM à la Crouzeix–Raviart for Highly Oscillatory Elliptic Problems, Chin. Ann. Math., Ser. B, Volume 34 (2013) no. 1, pp. 113-138 | DOI | MR | Zbl

[10] Claude Le Bris; Frédéric Legoll; François Madiot A numerical comparison of some Multiscale Finite Element approaches for advection-dominated problems in heterogeneous media, ESAIM, Math. Model. Numer. Anal., Volume 51 (2017) no. 3, pp. 851-888 | DOI | MR | Zbl

[11] Lam H. Nguyen; Dominik Schillinger A residual-driven local iterative corrector scheme for the multiscale finite element method, J. Comput. Phys., Volume 377 (2019), pp. 60-88 | DOI | MR | Zbl

[12] Rutger A. Biezemans; Claude Le Bris; Frédéric Legoll; Alexei Lozinski Non-intrusive implementation of Multiscale Finite Element Methods: An illustrative example, J. Comput. Phys., Volume 477 (2023), 111914 | DOI | MR | Zbl

[13] Pierre Grisvard Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, 24, Pitman Publishing Inc., 1985 | Zbl

[14] David Gilbarg; Neil S. Trudinger Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer, 2001 | DOI | Zbl

[15] Alfio Quarteroni Numerical Models for Differential Problems, MS&A. Modeling, Simulation and Applications, 16, Springer, 2018 | DOI | Zbl

[16] Philippe Gaston Ciarlet The finite element method for elliptic problems, Studies in Mathematics and its Applications, North-Holland, 1978 no. 4 | Zbl

[17] Alexandre Ern; Jean-Luc Guermond Theory and Practice of Finite Elements, Applied Mathematical Sciences, 159, Springer, 2004 | DOI | Zbl

[18] Alain Bensoussan; Jacques-Louis Lions; George Papanicolaou Asymptotic analysis for periodic structures, Studies in Mathematics and its Applications, 5, North-Holland, 1978 | Zbl

[19] Vasilij V. Zhikov; Sergej M. Kozlov; Olga A. Oleinik Homogenization of Differential Operators and Integral Functionals, Springer Berlin, 1994 | Zbl

[20] Grégoire Allaire Shape Optimization by the Homogenization Method, Applied Mathematical Sciences, 146, Springer, 2002 | DOI | Zbl

[21] François Murat; Luc Tartar $H$ -Convergence, Topics in the Mathematical Modelling of Composite Materials (Progress in Nonlinear Differential Equations and their Applications), Volume 31, Birkhäuser, 1997, pp. 21-43 | DOI | Zbl

[22] Rutger A. Biezemans Multiscale methods: non-intrusive implementation, advection-dominated problems and related topics, PhD thesis, École des Ponts ParisTech, Paris, France (2023) (in preparation.)

[23] Grégoire Allaire; Robert Brizzi A Multiscale Finite Element Method for Numerical Homogenization, Multiscale Model. Simul., Volume 4 (2005) no. 3, pp. 790-812 | DOI | MR | Zbl

[24] Jan S. Hesthaven; Shun Zhang; Xueyu Zhu High-Order Multiscale Finite Element Method for Elliptic Problems, Multiscale Model. Simul., Volume 12 (2014) no. 2, pp. 650-666 | DOI | MR | Zbl

[25] Frédéric Legoll; Pierre-Loïk Rothé; Claude Le Bris; Ulrich Hetmaniuk An MsFEM Approach Enriched Using Legendre Polynomials, Multiscale Model. Simul., Volume 20 (2022) no. 2, pp. 798-834 | DOI | MR | Zbl

[26] Peter Bastian; Markus Blatt; Andreas S. Dedner; Christian Engwer; Robert Klöfkorn; Mario Ohlberger; Oliver Sander A generic grid interface for parallel and adaptive scientific computing. Part I: abstract framework, Computing, Volume 82 (2008) no. 2-3, pp. 103-119 | DOI | Zbl

[27] Peter Bastian; Markus Blatt; Andreas S. Dedner; Christian Engwer; Robert Klöfkorn; Ralf Kornhuber; Mario Ohlberger; Oliver Sander A generic grid interface for parallel and adaptive scientific computing. Part II: implementation and tests in DUNE, Computing, Volume 82 (2008) no. 2-3, pp. 121-138 | DOI | Zbl

[28] Peter Bastian; Christian Engwer; Dominik Göddeke; Oleg Iliev; Olaf Ippisch; Mario Ohlberger; Stefan Turek; Jorrit Fahlke; Sven Kaulmann; Steffen Müthing; Dirk Ribbrock EXA-DUNE: Flexible PDE Solvers, Numerical Methods and Applications, Euro-Par 2014: Parallel Processing Workshops (Lecture Notes in Computer Science), Volume 8806, Springer, 2014, pp. 530-541 | DOI

[29] Peter Bastian; Christian Engwer; Jorrit Fahlke; Markus Geveler; Dominik Göddeke; Oleg Iliev; Olaf Ippisch; René Milk; Jan Mohring; Steffen Müthing; Mario Ohlberger; Dirk Ribbrock; Stefan Turek Advances Concerning Multiscale Methods and Uncertainty Quantification in EXA-DUNE, Software for Exascale Computing - SPPEXA 2013-2015 (Lecture Notes in Computational Science and Engineering), Volume 113, Springer, 2016, pp. 25-43 | DOI | MR

[30] Volker John; Joseph M. Maubach; Lutz Tobiska Nonconforming streamline-diffusion-finite-element-methods for convection-diffusion problems, Numer. Math., Volume 78 (1997) no. 2, pp. 165-188 | DOI | MR | Zbl

[31] Claude Le Bris; Frédéric Legoll; François Madiot Multiscale Finite Element Methods for Advection-Dominated Problems in Perforated Domains, Multiscale Model. Simul., Volume 17 (2019) no. 2, pp. 773-825 | DOI | MR | Zbl

[32] Claude Le Bris; Frédéric Legoll; Alexei Lozinski An MsFEM type approach for perforated domains, Multiscale Model. Simul., Volume 12 (2014) no. 3, pp. 1046-1077 | DOI | Zbl

[33] Pierre Degond; Alexei Lozinski; Bagus Putra Muljadi; Jacek Narski Crouzeix-Raviart MsFEM with Bubble Functions for Diffusion and Advection-Diffusion in Perforated Media, Commun. Comput. Phys., Volume 17 (2015) no. 4, pp. 887-907 | DOI | MR | Zbl

[34] Bagus Putra Muljadi; Jacek Narski; Alexei Lozinski; Pierre Degond Nonconforming Multiscale Finite Element Method for Stokes Flows in Heterogeneous Media. Part I: Methodologies and Numerical Experiments, Multiscale Model. Simul., Volume 13 (2015) no. 4, pp. 1146-1172 | DOI | MR | Zbl

[35] Gaspard Jankowiak; Alexei Lozinski Non-Conforming Multiscale Finite Element Method for Stokes Flows in Heterogeneous Media. Part II: error estimates for periodic microstructure (2018) (https://arxiv.org/abs/1802.04389, submitted)

[36] Thomas 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. Eng., Volume 127 (1995) no. 1-4, pp. 387-401 | DOI | MR | Zbl

[37] Thomas J. R. Hughes; Gonzalo R. Feijóo; Luca Mazzei; Jean-Baptiste Quincy The variational multiscale method—a paradigm for computational mechanics, Comput. Methods Appl. Mech. Eng., Volume 166 (1998) no. 1-2, pp. 3-24 | DOI | MR | Zbl

[38] Franco Brezzi; Alessandro Russo Choosing bubbles for advection-diffusion problems, Math. Models Methods Appl. Sci., Volume 4 (1994) no. 4, pp. 571-587 | DOI | MR | Zbl

[39] Franco Brezzi; L. P. Franca; Thomas J. R. Hughes; Alessandro Russo $b = \int g$ , Comput. Methods Appl. Mech. Eng., Volume 145 (1997), pp. 329-339 | Zbl

[40] Yalchin Efendiev; Thomas Y. Hou; Xiao-Hui Wu Convergence of a nonconforming multiscale finite element method, SIAM J. Numer. Anal., Volume 37 (2000) no. 3, pp. 888-910 | DOI | MR | Zbl

[41] Thomas Y. Hou; Xiao-Hui Wu; Yu Zhang Removing the Cell Resonance Error in the Multiscale Finite Element Method via a Petrov–Galerkin Formulation, Commun. Math. Sci., Volume 2 (2004) no. 2, pp. 185-205 | DOI | MR | Zbl

[42] Alexander N. Brooks; Thomas J. R. Hughes Streamline Upwind/Petrov–Galerkin Formulations for Convection Dominated Flow with Particular Emphasis on the Incompressible Navier–Stokes Equation, Comput. Methods Appl. Mech. Eng., Volume 32 (1982), pp. 199-259 | DOI | MR | Zbl

[43] Thomas J. R. Hughes; Leopoldo E. Franca; Gregory M. Hulbert A New Finite Element Method Formulation for Computational Fluid Dynamics: VIII. The Galerkin/Least-Squares Method for Advective-Diffusive Equations, Comput. Methods Appl. Mech. Eng., Volume 73 (1989), pp. 173-189 | DOI | MR | Zbl

[44] Leopoldo P. Franca; Alessandro Russo Recovering SUPG using Petrov–Galerkin formulations enriched with adjoint residual-free bubbles, Comput. Methods Appl. Mech. Eng., Volume 182 (2000) no. 3-4, pp. 333-339 | DOI | MR | Zbl

[45] Dietmar Gallistl; Daniel Peterseim Computation of Quasi-Local Effective Diffusion Tensors and Connections to the Mathematical Theory of Homogenization, Multiscale Model. Simul., Volume 15 (2017) no. 4, pp. 1530-1552 | DOI | MR | Zbl

[46] Daniel Elfverson; Victor Ginting; Patrick Henning On multiscale methods in Petrov–Galerkin formulation, Numer. Math., Volume 131 (2015) no. 4, pp. 643-682 | DOI | MR | Zbl

[47] Patrick Jenny; S. H. Lee; Hamdi A. Tchelepi Multi-scale finite-volume method for elliptic problems in subsurface flow simulation, J. Comput. Phys., Volume 187 (2003) no. 1, pp. 47-67 | DOI | Zbl

[48] Hadi Hajibeygi; Giuseppe Bonfigli; Marc Andre Hesse; Patrick Jenny Iterative multiscale finite-volume method, J. Comput. Phys., Volume 227 (2008) no. 19, pp. 8604-8621 | DOI | MR | Zbl

[49] Zhiming Chen; Thomas Y. Hou A mixed multiscale finite element method for elliptic problems with oscillating coefficients, Math. Comput., Volume 72 (2002) no. 242, pp. 541-576 | DOI | MR | Zbl

[50] Todd Arbogast Implementation of a Locally Conservative Numerical Subgrid Upscaling Scheme for Two-Phase Darcy Flow, Comput. Geosci., Volume 6 (2002), pp. 453-481 | DOI | MR | Zbl

[51] Todd Arbogast; Gergina Pencheva; Mary F. Wheeler; Ivan Yotov A Multiscale Mortar Mixed Finite Element Method, Multiscale Model. Simul., Volume 6 (2007) no. 1, pp. 319-346 | DOI | MR | Zbl

[52] Franco Brezzi; Jacques-Louis Lions; Olivier Pironneau The Chimera method for a model problem, Numerical Mathematics and Advanced Applications (Franco Brezzi; Annalisa Buffa; Stefania Corsaro; Almerico Murli, eds.), Springer, 2003, pp. 817-825 | DOI | Zbl

[53] Roland Glowinski; Jiwen He; Alexei Lozinski; Jacques Rappaz; Joël Wagner Finite element approximation of multi-scale elliptic problems using patches of elements, Numer. Math., Volume 101 (2005) no. 4, pp. 663-687 | DOI | MR | Zbl

[54] Jean-Baptiste Apoung Kamga; Olivier Pironneau Numerical zoom for multiscale problems with an application to nuclear waste disposal, J. Comput. Phys., Volume 224 (2007) no. 1, pp. 403-413 | DOI | MR | Zbl

[55] Mickaël Duval; Jean-Charles Passieux; Michel Salaün; Stéphane Guinard Non-intrusive Coupling: Recent Advances and Scalable Nonlinear Domain Decomposition, Arch. Comput. Methods Eng., Volume 23 (2016) no. 1, pp. 17-38 | DOI | MR | Zbl

[56] P. Gupta; J. P. Pereira; D.-J. Kim; C. A. Duarte; T. Eason Analysis of three-dimensional fracture mechanics problems: A non-intrusive approach using a generalized finite element method, Eng. Fract. Mech., Volume 90 (2012), pp. 41-64 | DOI

[57] Rachida Chakir; Yvon Maday Une méthode combinée d’éléments finis à deux grilles/bases réduites pour l’approximation des solutions d’une E.D.P. paramétrique (A two-grid finite-element/reduced basis scheme for the approximation of the solution of parametric dependent P.D.E.), C. R., Math., Acad. Sci. Paris, Volume 347 (2009) no. 7-8, pp. 435-440 | DOI | Zbl

[58] Rachida Chakir; Yvon Maday; Philippe Parnaudeau A non-intrusive reduced basis approach for parametrized heat transfer problems, J. Comput. Phys., Volume 376 (2019), pp. 617-633 | DOI | MR | Zbl

[59] Elise Grosjean; Yvon Maday Error estimate of the non-intrusive reduced basis method with finite volume schemes, ESAIM, Math. Model. Numer. Anal., Volume 55 (2021) no. 5, pp. 1941-1961 | DOI | MR | Zbl

[60] Claude Le Bris; Frédéric Legoll; Florian Thomines Multiscale Finite Element approach for “weakly” random problems and related issues, ESAIM, Math. Model. Numer. Anal., Volume 48 (2014) no. 3, pp. 815-858 | DOI | Numdam | MR | Zbl

[61] Frédéric Hecht New development in FreeFem++, J. Numer. Math., Volume 20 (2012) no. 3-4, pp. 251-265 | MR | Zbl

[62] Rutger A. Biezemans MsFEM in FreeFEM: Release version 1.0.0, 2023 (doi: 10.5281/zenodo.7525059)

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

Méthode des éléments finis multi-échelles pour le problème de Stokes

Alexei Lozinski; Zoubida Mghazli; Khallih Ould Ahmed Ould Blal

C. R. Math (2013)

Approximation grossière dʼun problème elliptique à coefficients hautement oscillants

Claude Le Bris; Frédéric Legoll; Kun Li

C. R. Math (2013)

Asymptotic Preserving numerical schemes for multiscale parabolic problems

Nicolas Crouseilles; Mohammed Lemou; Gilles Vilmart

C. R. Math (2016)