Comptes Rendus
Partial Differential Equations/Numerical Analysis
A new local reduced basis discontinuous Galerkin approach for heterogeneous multiscale problems
[Une nouvelle approche par bases réduites locale dʼune méthode de Galerkin discontinue pour des problèmes hétérogènes multi-échelles]
Comptes Rendus. Mathématique, Volume 349 (2011) no. 23-24, pp. 1233-1238.

Inspired by the reduced basis approach and modern numerical multiscale methods, we present a new framework for an efficient treatment of heterogeneous multiscale problems. The new approach is based on the idea of considering heterogeneous multiscale problems as parametrized partial differential equations where the parameters are smooth functions. We then construct, in an offline phase, a suitable localized reduced basis that is used in an online phase to efficiently compute approximations of the multiscale problem by means of a discontinuous Galerkin method on a coarse grid. We present our approach for elliptic multiscale problems and discuss an a posteriori error estimate that can be used in the construction process of the localized reduced basis. Numerical experiments are given to demonstrate the efficiency of the new approach.

Inspiré par lʼapproche des bases réduites et les méthodes numériques modernes pour des problèmes multi-échelles, nous présentons un nouveau traitement efficace des problèmes hétérogènes multi-échelles. La nouvelle approche repose sur lʼidée de considérer des problèmes hétérogènes multi-échelles comme des équations différentielles partielles paramétrisées, où les paramètres sont des fonctions lisses. Nous construisons alors dans une phase « offline » une base réduite localisée appropriée, utilisée dans une phase « online » pour calculer efficacement des approximations du problème multi-échelle par une méthode Galerkin discontinue sur un maillage grossier. Nous présentons notre nouvelle approche pour des problèmes elliptiques multi-échelles et discutons une estimation dʼerreur à posteriori utilisée lors de la construction de la base réduite localisée. Des expériences numériques sont exposées pour démontrer la efficacité de la nouvelle approche.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2011.10.024

Sven Kaulmann 1 ; Mario Ohlberger 2 ; Bernard Haasdonk 1

1 Institute of Applied Analysis and Numerical Simulation, University of Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
2 Institute of Computational and Applied Mathematics & Center for Nonlinear Science, University of Münster, Einsteinstr. 62, 48149 Münster, Germany
@article{CRMATH_2011__349_23-24_1233_0,
     author = {Sven Kaulmann and Mario Ohlberger and Bernard Haasdonk},
     title = {A new local reduced basis discontinuous {Galerkin} approach for heterogeneous multiscale problems},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1233--1238},
     publisher = {Elsevier},
     volume = {349},
     number = {23-24},
     year = {2011},
     doi = {10.1016/j.crma.2011.10.024},
     language = {en},
}
TY  - JOUR
AU  - Sven Kaulmann
AU  - Mario Ohlberger
AU  - Bernard Haasdonk
TI  - A new local reduced basis discontinuous Galerkin approach for heterogeneous multiscale problems
JO  - Comptes Rendus. Mathématique
PY  - 2011
SP  - 1233
EP  - 1238
VL  - 349
IS  - 23-24
PB  - Elsevier
DO  - 10.1016/j.crma.2011.10.024
LA  - en
ID  - CRMATH_2011__349_23-24_1233_0
ER  - 
%0 Journal Article
%A Sven Kaulmann
%A Mario Ohlberger
%A Bernard Haasdonk
%T A new local reduced basis discontinuous Galerkin approach for heterogeneous multiscale problems
%J Comptes Rendus. Mathématique
%D 2011
%P 1233-1238
%V 349
%N 23-24
%I Elsevier
%R 10.1016/j.crma.2011.10.024
%G en
%F CRMATH_2011__349_23-24_1233_0
Sven Kaulmann; Mario Ohlberger; Bernard Haasdonk. A new local reduced basis discontinuous Galerkin approach for heterogeneous multiscale problems. Comptes Rendus. Mathématique, Volume 349 (2011) no. 23-24, pp. 1233-1238. doi : 10.1016/j.crma.2011.10.024. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.10.024/

[1] J.E. Aarnes; Y. Efendiev; L. Jiang Mixed multiscale finite element methods using limited global information, Multiscale Model. Simul., Volume 7 (2008) no. 2, pp. 655-676

[2] D.N. Arnold; F. Brezzi; B. Cockburn; L.D. Marini Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., Volume 39 (2002) no. 5, pp. 1749-1779

[3] Y. Chen; J.S. Hesthaven; Y. Maday A seamless reduced basis element method for 2D Maxwellʼs problem: An introduction, Spectral and High Order Methods for Partial Differential Equations, Selected Papers from the ICOSAHOMʼ09 Conference, vol. 76, 2011, pp. 141-152

[4] I.T. Jolliffe Principal Component Analysis, John Wiley & Sons, 2002

[5] Sven Kaulmann, A localized reduced basis approach for heterogeneous multiscale problems, Diploma thesis, University of Münster, 2011, available online: http://www.agh.ians.uni-stuttgart.de/orga/people/kaulmann.html.

[6] A.E. Løvgren; Y. Maday; E.M. Rønquist A reduced basis element method for the steady Stokes problem, M2AN Math. Model. Numer. Anal., Volume 40 (2006), pp. 529-552

[7] Y. Maday; E.M. Rønquist The reduced basis element method: application to a thermal fin problem, SIAM J. Sci. Comput., Volume 26 (2004) no. 1, pp. 240-258 (electronic)

[8] M. Ohlberger, Wissenschaftliches Rechnen, Lecture Notes, University of Münster, 2010.

  • Francesco Romor; Davide Torlo; Gianluigi Rozza Friedrichs' systems discretized with the DGM: domain decomposable model order reduction and Graph Neural Networks approximating vanishing viscosity solutions, Journal of Computational Physics, Volume 531 (2025), p. 113915 | DOI:10.1016/j.jcp.2025.113915
  • Mehran Ebrahimi; Masayuki Yano A hyperreduced reduced basis element method for reduced-order modeling of component-based nonlinear systems, Computer Methods in Applied Mechanics and Engineering, Volume 431 (2024), p. 117254 | DOI:10.1016/j.cma.2024.117254
  • Ewann Gautier; Bruno Levieil; Sylvain Calloch; Cédric Doudard A new adjustable localization operator method to compute local stresses and strains at notched bodies under multiaxial cyclic loading, International Journal of Fatigue, Volume 182 (2024), p. 108194 | DOI:10.1016/j.ijfatigue.2024.108194
  • Francesca Bonizzoni; Moritz Hauck; Daniel Peterseim A reduced basis super-localized orthogonal decomposition for reaction-convection-diffusion problems, Journal of Computational Physics, Volume 499 (2024), p. 112698 | DOI:10.1016/j.jcp.2023.112698
  • Angelo Iollo; Giulia Sambataro; Tommaso Taddei A one-shot overlapping Schwarz method for component-based model reduction: application to nonlinear elasticity, Computer Methods in Applied Mechanics and Engineering, Volume 404 (2023), p. 115786 | DOI:10.1016/j.cma.2022.115786
  • A. Giuliodori; J.A. Hernández; E. Soudah Multiscale modeling of prismatic heterogeneous structures based on a localized hyperreduced-order method, Computer Methods in Applied Mechanics and Engineering, Volume 407 (2023), p. 115913 | DOI:10.1016/j.cma.2023.115913
  • Stein K.F. Stoter; Bernardo Cockburn; Thomas J.R. Hughes; Dominik Schillinger Discontinuous Galerkin methods through the lens of variational multiscale analysis, Computer Methods in Applied Mechanics and Engineering, Volume 388 (2022), p. 114220 | DOI:10.1016/j.cma.2021.114220
  • Angelo Iollo; Giulia Sambataro; Tommaso Taddei An adaptive projection‐based model reduction method for nonlinear mechanics with internal variables: Application to thermo‐hydro‐mechanical systems, International Journal for Numerical Methods in Engineering, Volume 123 (2022) no. 12, p. 2894 | DOI:10.1002/nme.6964
  • Chi Hoang; Youngsoo Choi; Kevin Carlberg Domain-decomposition least-squares Petrov–Galerkin (DD-LSPG) nonlinear model reduction, Computer Methods in Applied Mechanics and Engineering, Volume 384 (2021), p. 113997 | DOI:10.1016/j.cma.2021.113997
  • Laurent Lacourt; David Ryckelynck; Samuel Forest; Victor de Rancourt; Sylvain Flouriot Hyper‐reduced direct numerical simulation of voids in welded joints via image‐based modeling, International Journal for Numerical Methods in Engineering, Volume 121 (2020) no. 11, p. 2581 | DOI:10.1002/nme.6320
  • P.S.B. Nigro; E.T. Simões; P.M. Pimenta; J. Schröder Model order reduction with Galerkin projection applied to nonlinear optimization with infeasible primal‐dual interior point method, International Journal for Numerical Methods in Engineering, Volume 120 (2019) no. 12, p. 1310 | DOI:10.1002/nme.6181
  • Andrea Ferrero; Angelo Iollo; Francesco Larocca Global and local POD models for the prediction of compressible flows with DG methods, International Journal for Numerical Methods in Engineering, Volume 116 (2018) no. 5, p. 332 | DOI:10.1002/nme.5927
  • Tomáš Dohnal; Ben Schweizer A Bloch Wave Numerical Scheme for Scattering Problems in Periodic Wave-Guides, SIAM Journal on Numerical Analysis, Volume 56 (2018) no. 3, p. 1848 | DOI:10.1137/17m1141643
  • Mario Ohlberger; Michael Schaefer; Felix Schindler Localized Model Reduction in PDE Constrained Optimization, Shape Optimization, Homogenization and Optimal Control, Volume 169 (2018), p. 143 | DOI:10.1007/978-3-319-90469-6_8
  • Mario Ohlberger; Felix Schindler Non-conforming Localized Model Reduction with Online Enrichment: Towards Optimal Complexity in PDE Constrained Optimization, Finite Volumes for Complex Applications VIII - Hyperbolic, Elliptic and Parabolic Problems, Volume 200 (2017), p. 357 | DOI:10.1007/978-3-319-57394-6_38
  • Daniel S. Meyer; Brian T. Helenbrook; Ming‐C. Cheng Proper orthogonal decomposition‐based reduced basis element thermal modeling of integrated circuits, International Journal for Numerical Methods in Engineering, Volume 112 (2017) no. 5, p. 479 | DOI:10.1002/nme.5529
  • Filippo Salmoiraghi; Francesco Ballarin; Luca Heltai; Gianluigi Rozza Isogeometric analysis-based reduced order modelling for incompressible linear viscous flows in parametrized shapes, Advanced Modeling and Simulation in Engineering Sciences, Volume 3 (2016) no. 1 | DOI:10.1186/s40323-016-0076-6
  • Jan S. Hesthaven; Gianluigi Rozza; Benjamin Stamm Introduction, Certified Reduced Basis Methods for Parametrized Partial Differential Equations (2016), p. 1 | DOI:10.1007/978-3-319-22470-1_1
  • Paolo Pacciarini; Paola Gervasio; Alfio Quarteroni Spectral based Discontinuous Galerkin Reduced Basis Element method for parametrized Stokes problems, Computers Mathematics with Applications, Volume 72 (2016) no. 8, p. 1977 | DOI:10.1016/j.camwa.2016.01.030
  • Paola F. Antonietti; Paolo Pacciarini; Alfio Quarteroni A discontinuous Galerkin reduced basis element method for elliptic problems, ESAIM: Mathematical Modelling and Numerical Analysis, Volume 50 (2016) no. 2, p. 337 | DOI:10.1051/m2an/2015045
  • Kristin Kirchner; Karsten Urban; Oliver Zeeb Maxwell’s equations for conductors with impedance boundary conditions: Discontinuous Galerkin and Reduced Basis Methods, ESAIM: Mathematical Modelling and Numerical Analysis, Volume 50 (2016) no. 6, p. 1763 | DOI:10.1051/m2an/2016006
  • M. Redeker; B. Haasdonk A POD–EIM reduced two-scale model for precipitation in porous media, Mathematical and Computer Modelling of Dynamical Systems, Volume 22 (2016) no. 4, p. 323 | DOI:10.1080/13873954.2016.1198384
  • Mario Ohlberger; Stephan Rave; Felix Schindler Model Reduction for Multiscale Lithium-Ion Battery Simulation, Numerical Mathematics and Advanced Applications ENUMATH 2015, Volume 112 (2016), p. 317 | DOI:10.1007/978-3-319-39929-4_31
  • Assyr Abdulle; Yun Bai; Gilles Vilmart Reduced basis finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems, Discrete Continuous Dynamical Systems - S, Volume 8 (2015) no. 1, p. 91 | DOI:10.3934/dcdss.2015.8.91
  • S. Kaulmann; B. Flemisch; B. Haasdonk; K. ‐A. Lie; M. Ohlberger The localized reduced basis multiscale method for two‐phase flows in porous media, International Journal for Numerical Methods in Engineering, Volume 102 (2015) no. 5, p. 1018 | DOI:10.1002/nme.4773
  • Assyr Abdulle; Patrick Henning A reduced basis localized orthogonal decomposition, Journal of Computational Physics, Volume 295 (2015), p. 379 | DOI:10.1016/j.jcp.2015.04.016
  • Jan S. Hesthaven; Shun Zhang; Xueyu Zhu Reduced Basis Multiscale Finite Element Methods for Elliptic Problems, Multiscale Modeling Simulation, Volume 13 (2015) no. 1, p. 316 | DOI:10.1137/140955070
  • M. Ohlberger; F. Schindler Error Control for the Localized Reduced Basis Multiscale Method with Adaptive On-Line Enrichment, SIAM Journal on Scientific Computing, Volume 37 (2015) no. 6, p. A2865 | DOI:10.1137/151003660
  • Y. Efendiev; J. Galvis; R. Lazarov; M. Moon; M. Sarkis Generalized multiscale finite element method. Symmetric interior penalty coupling, Journal of Computational Physics, Volume 255 (2013), p. 1 | DOI:10.1016/j.jcp.2013.07.028
  • Sven Kaulmann; Mario Ohlberger; Bernard Haasdonk A new local reduced basis discontinuous Galerkin approach for heterogeneous multiscale problems, Comptes Rendus. Mathématique, Volume 349 (2011) no. 23-24, p. 1233 | DOI:10.1016/j.crma.2011.10.024

Cité par 30 documents. Sources : Crossref

Commentaires - Politique