Skip to main content
    • Aa
    • Aa
  • Get access
    Check if you have access via personal or institutional login
  • Cited by 46
  • Cited by
    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Abdulle, Assyr and Budáč, Ondrej 2016. A reduced basis finite element heterogeneous multiscale method for Stokes flow in porous media. Computer Methods in Applied Mechanics and Engineering, Vol. 307, p. 1.

    Abdulle, Assyr and Pouchon, Timothée 2016. A Priori Error Analysis of the Finite Element Heterogeneous Multiscale Method for the Wave Equation over Long Time. SIAM Journal on Numerical Analysis, Vol. 54, Issue. 3, p. 1507.

    Brown, Donald L. and Vasilyeva, Maria 2016. A generalized multiscale finite element method for poroelasticity problems II: Nonlinear coupling. Journal of Computational and Applied Mathematics, Vol. 297, p. 132.

    Brown, Donald L. and Vasilyeva, Maria 2016. A Generalized Multiscale Finite Element Method for poroelasticity problems I: Linear problems. Journal of Computational and Applied Mathematics, Vol. 294, p. 372.

    Cao, Meng and Roberts, A. J. 2016. Multiscale modelling couples patches of non-linear wave-like simulations. IMA Journal of Applied Mathematics, Vol. 81, Issue. 2, p. 228.

    Condon, Marissa Gao, Jing and Iserles, Arieh 2016. On asymptotic expansion solvers for highly oscillatory semi-explicit DAEs. Discrete and Continuous Dynamical Systems, Vol. 36, Issue. 9, p. 4813.

    Crouseilles, Nicolas Lemou, Mohammed and Vilmart, Gilles 2016. Asymptotic Preserving numerical schemes for multiscale parabolic problems. Comptes Rendus Mathematique, Vol. 354, Issue. 3, p. 271.

    Lee, Yoonsang and Engquist, Bjorn 2016. Multiscale numerical methods for passive advection–diffusion in incompressible turbulent flow fields. Journal of Computational Physics, Vol. 317, p. 33.

    Ma, Qiang Cui, Junzhi Li, Zhihui and Wang, Ziqiang 2016. Second-order asymptotic algorithm for heat conduction problems of periodic composite materials in curvilinear coordinates. Journal of Computational and Applied Mathematics, Vol. 306, p. 87.

    Shekhar, Prashant Patra, Abani and Stefanescu, E.R. 2016. Multilevel Methods for Sparse Representation of Topographical Data. Procedia Computer Science, Vol. 80, p. 887.

    Abdulle, A. Huber, M. E. and Vilmart, G. 2015. Linearized Numerical Homogenization Method for Nonlinear Monotone Parabolic Multiscale Problems. Multiscale Modeling & Simulation, Vol. 13, Issue. 3, p. 916.

    Bunder, J.E. and Roberts, A.J. 2015. Numerical integration of ordinary differential equations with rapidly oscillatory factors. Journal of Computational and Applied Mathematics, Vol. 282, p. 54.

    Cao, Wanrong Zhang, Zhongqiang and Karniadakis, George Em 2015. Time-Splitting Schemes for Fractional Differential Equations I: Smooth Solutions. SIAM Journal on Scientific Computing, Vol. 37, Issue. 4, p. A1752.

    Elfverson, Daniel Ginting, Victor and Henning, Patrick 2015. On multiscale methods in Petrov–Galerkin formulation. Numerische Mathematik, Vol. 131, Issue. 4, p. 643.

    Fafalis, Dimitrios and Fish, Jacob 2015. Computational aspects of dispersive computational continua for elastic heterogeneous media. Computational Mechanics, Vol. 56, Issue. 6, p. 931.

    Geiser, Jürgen 2015. Recent Advances in Splitting Methods for Multiphysics and Multiscale: Theory and Applications. Journal of Algorithms & Computational Technology, Vol. 9, Issue. 1, p. 65.

    Henning, Patrick Ohlberger, Mario and Schweizer, Ben 2015. Adaptive heterogeneous multiscale methods for immiscible two-phase flow in porous media. Computational Geosciences, Vol. 19, Issue. 1, p. 99.

    Jones, C. R. Bretherton, C. S. and Pritchard, M. S. 2015. Mean-state acceleration of cloud-resolving models and large eddy simulations. Journal of Advances in Modeling Earth Systems, Vol. 7, Issue. 4, p. 1643.

    Liu, Ping Samaey, Giovanni William Gear, C. and Kevrekidis, Ioannis G. 2015. On the acceleration of spatially distributed agent-based computations: A patch dynamics scheme. Applied Numerical Mathematics, Vol. 92, p. 54.

    Maclean, John 2015. A Note on Implementations of the Boosting Algorithm and Heterogeneous Multiscale Methods. SIAM Journal on Numerical Analysis, Vol. 53, Issue. 5, p. 2472.


The heterogeneous multiscale method*

  • Assyr Abdulle (a1), E Weinan (a2), Björn Engquist (a3) and Eric Vanden-Eijnden (a4)
  • DOI:
  • Published online: 01 April 2012

The heterogeneous multiscale method (HMM), a general framework for designing multiscale algorithms, is reviewed. Emphasis is given to the error analysis that comes naturally with the framework. Examples of finite element and finite difference HMM are presented. Applications to dynamical systems and stochastic simulation algorithms with multiple time scales, spall fracture and heat conduction in microprocessors are discussed.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Acta Numerica
  • ISSN: 0962-4929
  • EISSN: 1474-0508
  • URL: /core/journals/acta-numerica
Please enter your name
Please enter a valid email address
Who would you like to send this to? *