February  2015, 8(1): i-vi. doi: 10.3934/dcdss.2015.8.1i

An attempt at classifying homogenization-based numerical methods

1. 

Université de Bretagne-Sud, UMR 6205, LMBA, F-56000 Vannes, France

Published  July 2014

In this note, a classification of Homogenization-Based Numerical Methods and (in particular) of Numerical Methods that are based on the Two-Scale Convergence is done. In this classification stand: Direct Homogenization-Based Numerical Methods; H-Measure-Based Numerical Methods; Two-Scale Numerical Methods and TSAPS: Two-Scale Asymptotic Preserving Schemes.
Citation: 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
References:
[1]

A. Abdulle, Y. Bai and G. Vilmart, Reduced basis finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems,, Discrete and Continuous Dynamical Systems - Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence, (). 

[2]

A. Back and E. Frénod, Geometric two-scale convergence on manifold and applications to the Vlasov equation,, Discrete and Continuous Dynamical Systems - Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence, (). 

[3]

J.-P. Bernard, E. Frénod and A. Rousseau, Paralic confinement computations in coastal environment with interlocked areas,, Discrete and Continuous Dynamical Systems - Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence, (). 

[4]

N. Crouseilles, E. Frenod, S. Hirstoaga and A. Mouton, Two-Scale Macro-Micro decomposition of the Vlasov equation with a strong magnetic field, Mathematical Models and Methods in Applied Sciences, 23 (2013), 1527-1559. doi: 10.1142/S0218202513500152.

[5]

I. Faye, E. Frénod and D. Seck, Two-scale numerical simulation of sand transport problems,, Discrete and Continuous Dynamical Systems - Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence, (). 

[6]

E. Frénod, S. Histoaga and E. Sonnendrücker, An exponential integrator for a highly oscillatory Vlasov equation,, Discrete and Continuous Dynamical Systems - Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence, (). 

[7]

P. Henning and M. Ohlberger, Error control and adaptivity for heterogeneous multiscale approximations of nonlinear monotone problems,, Discrete and Continuous Dynamical Systems - Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence, (). 

[8]

S. Jin, Efficient asymptotic-preserving (ap) schemes for some multiscale kinetic equations, SIAM Journal of Scientific Computing, 21 (1999), 441-454. doi: 10.1137/S1064827598334599.

[9]

V. Laptev, Deterministic homogenization for media with barriers,, Discrete and Continuous Dynamical Systems - Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence, (). 

[10]

F. Legoll and W. Minvielle, Variance reduction using antithetic variables for a nonlinear convex stochastic homogenization problem,, Discrete and Continuous Dynamical Systems - Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence, (). 

[11]

M. Lutz, Application of Lie transform techniques for simulation of a charged particle beam,, Discrete and Continuous Dynamical Systems - Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence, (). 

[12]

Tartar, Multi-scales h-measures,, Discrete and Continuous Dynamical Systems - Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence, (). 

[13]

X. Xu and S. Yue, Homogenization of thermal-hydro-mass transfer processes,, Discrete and Continuous Dynamical Systems - Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence, (). 

show all references

References:
[1]

A. Abdulle, Y. Bai and G. Vilmart, Reduced basis finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems,, Discrete and Continuous Dynamical Systems - Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence, (). 

[2]

A. Back and E. Frénod, Geometric two-scale convergence on manifold and applications to the Vlasov equation,, Discrete and Continuous Dynamical Systems - Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence, (). 

[3]

J.-P. Bernard, E. Frénod and A. Rousseau, Paralic confinement computations in coastal environment with interlocked areas,, Discrete and Continuous Dynamical Systems - Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence, (). 

[4]

N. Crouseilles, E. Frenod, S. Hirstoaga and A. Mouton, Two-Scale Macro-Micro decomposition of the Vlasov equation with a strong magnetic field, Mathematical Models and Methods in Applied Sciences, 23 (2013), 1527-1559. doi: 10.1142/S0218202513500152.

[5]

I. Faye, E. Frénod and D. Seck, Two-scale numerical simulation of sand transport problems,, Discrete and Continuous Dynamical Systems - Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence, (). 

[6]

E. Frénod, S. Histoaga and E. Sonnendrücker, An exponential integrator for a highly oscillatory Vlasov equation,, Discrete and Continuous Dynamical Systems - Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence, (). 

[7]

P. Henning and M. Ohlberger, Error control and adaptivity for heterogeneous multiscale approximations of nonlinear monotone problems,, Discrete and Continuous Dynamical Systems - Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence, (). 

[8]

S. Jin, Efficient asymptotic-preserving (ap) schemes for some multiscale kinetic equations, SIAM Journal of Scientific Computing, 21 (1999), 441-454. doi: 10.1137/S1064827598334599.

[9]

V. Laptev, Deterministic homogenization for media with barriers,, Discrete and Continuous Dynamical Systems - Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence, (). 

[10]

F. Legoll and W. Minvielle, Variance reduction using antithetic variables for a nonlinear convex stochastic homogenization problem,, Discrete and Continuous Dynamical Systems - Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence, (). 

[11]

M. Lutz, Application of Lie transform techniques for simulation of a charged particle beam,, Discrete and Continuous Dynamical Systems - Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence, (). 

[12]

Tartar, Multi-scales h-measures,, Discrete and Continuous Dynamical Systems - Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence, (). 

[13]

X. Xu and S. Yue, Homogenization of thermal-hydro-mass transfer processes,, Discrete and Continuous Dynamical Systems - Serie S. Special Issue on Numerical Methods based on Homogenization and Two-Scale Convergence, (). 

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