Multiscale stochastic homogenization of convection-diffusion equations

Nils Svanstedt

Applications of Mathematics (2008)

  • Volume: 53, Issue: 2, page 143-155
  • ISSN: 0862-7940

Abstract

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Multiscale stochastic homogenization is studied for convection-diffusion problems. More specifically, we consider the asymptotic behaviour of a sequence of realizations of the form ∂ u ε ω / ∂ t + 1 / ϵ 3 𝒞 T 3 ( x / ε 3 ) ω 3 · ∇ u ε ω - div α T 1 ( x / ε 1 ) ω 1 , T 2 ( x / ε 2 ) ω 2 , t ∇ u ε ω = f . It is shown, under certain structure assumptions on the random vector field 𝒞 ( ω 3 ) and the random map α ( ω 1 , ω 2 , t ) , that the sequence { u ϵ ω } of solutions converges in the sense of G-convergence of parabolic operators to the solution u of the homogenized problem ∂ u / ∂ t - div ( ℬ ( t ) ∇ u ) = f .

How to cite

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Svanstedt, Nils. "Multiscale stochastic homogenization of convection-diffusion equations." Applications of Mathematics 53.2 (2008): 143-155. <http://eudml.org/doc/33315>.

@article{Svanstedt2008,
abstract = {Multiscale stochastic homogenization is studied for convection-diffusion problems. More specifically, we consider the asymptotic behaviour of a sequence of realizations of the form $\{\partial u^\omega _\{\varepsilon \}\}/\{\partial t\} +\{1\}/\{\epsilon _3\}\,\mathcal \{C\}\bigl (T_3(\{x\}/\{\varepsilon _3\}) \omega _3\bigr )\cdot \nabla u^\omega _\{\varepsilon \}- \operatorname\{div\}\bigl ( \alpha \bigl (T_1(\{x\}/\{\varepsilon _1\})\omega _1, T_2(\{x\}/\{\varepsilon _2\})\omega _2 ,t\bigr ) \nabla u^\omega _\{\varepsilon \}\bigr )=f$. It is shown, under certain structure assumptions on the random vector field $\{\mathcal \{C\}\}(\omega _3)$ and the random map $\alpha (\omega _1,\omega _2,t)$, that the sequence $\lbrace u^\omega _\epsilon \rbrace $ of solutions converges in the sense of G-convergence of parabolic operators to the solution $u$ of the homogenized problem $\{\partial u\}/\{\partial t\} - \operatorname\{div\}( \mathcal \{B\}(t)\nabla u ) = f$.},
author = {Svanstedt, Nils},
journal = {Applications of Mathematics},
keywords = {multiscale; stochastic; homogenization; convection-diffusion; multiscale; stochastic; homogenization; convection-diffusion},
language = {eng},
number = {2},
pages = {143-155},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Multiscale stochastic homogenization of convection-diffusion equations},
url = {http://eudml.org/doc/33315},
volume = {53},
year = {2008},
}

TY - JOUR
AU - Svanstedt, Nils
TI - Multiscale stochastic homogenization of convection-diffusion equations
JO - Applications of Mathematics
PY - 2008
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 53
IS - 2
SP - 143
EP - 155
AB - Multiscale stochastic homogenization is studied for convection-diffusion problems. More specifically, we consider the asymptotic behaviour of a sequence of realizations of the form ${\partial u^\omega _{\varepsilon }}/{\partial t} +{1}/{\epsilon _3}\,\mathcal {C}\bigl (T_3({x}/{\varepsilon _3}) \omega _3\bigr )\cdot \nabla u^\omega _{\varepsilon }- \operatorname{div}\bigl ( \alpha \bigl (T_1({x}/{\varepsilon _1})\omega _1, T_2({x}/{\varepsilon _2})\omega _2 ,t\bigr ) \nabla u^\omega _{\varepsilon }\bigr )=f$. It is shown, under certain structure assumptions on the random vector field ${\mathcal {C}}(\omega _3)$ and the random map $\alpha (\omega _1,\omega _2,t)$, that the sequence $\lbrace u^\omega _\epsilon \rbrace $ of solutions converges in the sense of G-convergence of parabolic operators to the solution $u$ of the homogenized problem ${\partial u}/{\partial t} - \operatorname{div}( \mathcal {B}(t)\nabla u ) = f$.
LA - eng
KW - multiscale; stochastic; homogenization; convection-diffusion; multiscale; stochastic; homogenization; convection-diffusion
UR - http://eudml.org/doc/33315
ER -

References

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  9. G-convergence and homogenization of sequences of linear and monlinear partial differential operators, PhD. Thesis, Luleå University, 1992. (1992) 
  10. 10.1016/S0362-546X(97)00532-4, Nonlinear Anal., Theory Methods Appl. 36 (1999), 807–842. (1999) Zbl0933.35020MR1682689DOI10.1016/S0362-546X(97)00532-4
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