Unfolding-based corrector estimates for a reaction–diffusion system predicting concrete corrosion

Fatima, Tasnim; Muntean, Adrian; Ptashnyk, Mariya

doi:10.1080/00036811.2011.625016

Cited by 23 publications

(24 citation statements)

References 13 publications

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“…In this new geometrical setting, the corresponding compactness results (see, for instance, [] and []) imply that there exist

u_{1} \in H_{0}^{1} false(normalΩ false)

{\overset{u}{true}}_{1} \in L^{2} false(normalΩ, H_{per}^{1} (Y_{1}) false)

u_{2} \in H_{0}^{1} false(normalΩ false)

{\bar{u}}_{2} \in L^{2} false(normalΩ, H_{per}^{1} (Y_{2}) false)

, such that

{scriptM}_{normalΓ} false({\hat{u}}_{1} false) = 0

{scriptM}_{normalΓ} false({\bar{u}}_{2} false) = 0

and, up to a subsequence, for

ε \to 0

, we have:

\begin{matrix} left {scriptT}_{1}^{ε} (u_{1}^{ε}) \to u_{1} strongly in L^{2} (Ω, H^{1} false(Y_{1} false)), \\ left {scriptT}_{1}^{ε} (\nabla u_{1}^{ε}) ⇀ \nabla u_{1} + \nabla_{y} {\hat{u}}_{1} weakly in L^{2} (Ω \times Y_{1}), \\ left {scriptT}_{2}^{ε} (u_{2}^{ε}) \to u_{2} strongly in L^{2} (Ω, H^{1} ... \end{matrix}

…”

Section: The Connected–connected Case; a Comparison Resultsmentioning

confidence: 99%

“…The variational formulation of problem (43) is exactly (8) written for this new space endowed with the corresponding scalar product (6). In this new geometrical setting, the corresponding compactness results (see, for instance, [24] and [32]) imply that there exist 1 ∈ 1 0 (Ω),̂1 ∈ 2 (Ω, 1 per ( 1 )), 2 ∈ 1 0 (Ω), 2 ∈ 2 (Ω, 1 per ( 2 )), such that  Γ (̂1) = 0,  Γ ( 2 ) = 0 and, up to a subsequence, for → 0, we have:  1 ( 1 ) → 1 strongly in 2 (Ω, 1 ( 1 )),  1 (∇ 1 ) ⇀ ∇ 1 + ∇̂1 weakly in 2 (Ω × 1 ),  2 ( 2 ) → 2 strongly in 2 (Ω, 1 ( 2 )),  2 (∇ 2 ) ⇀ ∇ 2 + ∇ 2 weakly in 2 (Ω × 2 ),…”

Section: The Connected-connected Case; a Comparison Resultsmentioning

confidence: 99%

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Upscaling of a diffusion problem with interfacial flux jump leading to a modified Barenblatt model

Bunoiu

Timofte

2018

Z Angew Math Mech

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In this paper, we study the homogenization of a diffusion problem in a highly heterogeneous composite medium formed by two constituents separated by an imperfect interface, where both the temperature and the flux exhibit jumps. The presence of the flux jump leads to a modified stationary Barenblatt model. Besides, we discuss two different geometrical settings, providing a mathematical justification for a physical phenomenon which is explained by the connectivity properties of the constituents.

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“…In this new geometrical setting, the corresponding compactness results (see, for instance, [] and []) imply that there exist

u_{1} \in H_{0}^{1} false(normalΩ false)

{\overset{u}{true}}_{1} \in L^{2} false(normalΩ, H_{per}^{1} (Y_{1}) false)

u_{2} \in H_{0}^{1} false(normalΩ false)

{\bar{u}}_{2} \in L^{2} false(normalΩ, H_{per}^{1} (Y_{2}) false)

, such that

{scriptM}_{normalΓ} false({\hat{u}}_{1} false) = 0

{scriptM}_{normalΓ} false({\bar{u}}_{2} false) = 0

and, up to a subsequence, for

ε \to 0

, we have:

\begin{matrix} left {scriptT}_{1}^{ε} (u_{1}^{ε}) \to u_{1} strongly in L^{2} (Ω, H^{1} false(Y_{1} false)), \\ left {scriptT}_{1}^{ε} (\nabla u_{1}^{ε}) ⇀ \nabla u_{1} + \nabla_{y} {\hat{u}}_{1} weakly in L^{2} (Ω \times Y_{1}), \\ left {scriptT}_{2}^{ε} (u_{2}^{ε}) \to u_{2} strongly in L^{2} (Ω, H^{1} ... \end{matrix}

…”

Section: The Connected–connected Case; a Comparison Resultsmentioning

confidence: 99%

Section: The Connected-connected Case; a Comparison Resultsmentioning

confidence: 99%

Upscaling of a diffusion problem with interfacial flux jump leading to a modified Barenblatt model

Bunoiu

Timofte

2018

Z Angew Math Mech

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“…In [20], the main working tools involve the two-scale convergence concept in the sense of Nguetseng and Allaire 1 combined with a periodic unfolding of the oscillatory boundary. For that reaction-diffusion scenario, we have used the periodic unfolding technique to get correctors very much in the spirit of Griso [21]. Here the situation we are looking at is conceptually different -the microstructures are now allowed to be distributed in a locally (non-uniform!)…”

Section: Introductionmentioning

confidence: 99%

Corrector estimates for the homogenization of a locally periodic medium with areas of low and high diffusivity

Muntean

Noorden

2013

Eur. J. Appl. Math

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We prove an upper bound for the convergence rate of the homogenization limit → 0 for a linear transmission problem for a advection-diffusion(-reaction) system posed in areas with low and high diffusivity, where is a suitable scale parameter. In this way we rigorously justify the formal homogenization asymptotics obtained in [37] (van Noorden, T. and Muntean, A. (2011) Homogenization of a locally-periodic medium with areas of low and high diffusivity. Eur. J. Appl. Math. 22, 493-516). We do this by providing a corrector estimate. The main ingredients for the proof of the correctors include integral estimates for rapidly oscillating functions with prescribed average, properties of the macroscopic reconstruction operators, energy bounds, and extra two-scale regularity estimates. The whole procedure essentially relies on a good understanding of the analysis of the limit two-scale problem.

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“…Based on this strong notion of convergence, one can ask for quantitative error estimates, see e.g. [17,18,19,20,21], as well as for numerical simulations, see e.g. [22,23,24,25,26,27].…”

Section: Introductionmentioning

confidence: 99%

“…We assume neither additional spatial regularity of the original solutions (u ε , v ε ) nor of the corrector functions. In [20], a reaction-diffusion system predicting concrete corrosion is considered, but the system does not include slowly diffusing species v ε . Nevertheless, for the classically diffusing species u ε and its gradient ∇u ε the convergence rate ε 1/2 and ε 1/4 , respectively, is rigorously proved by the method of periodic unfolding.…”

Section: Introductionmentioning

confidence: 99%

Error estimates for nonlinear reaction-diffusion systems involving different diffusion length scales

Reichelt

2016

J. Phys.: Conf. Ser.

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Abstract. We derive quantitative error estimates for coupled reaction-diffusion systems, whose coefficient functions are quasi-periodically oscillating modeling the microstructure of the underlying macroscopic domain. The coupling arises via nonlinear reaction terms and we allow for different diffusion length scales, i.e. whereas some species have characteristic diffusion length of order 1 other species may diffuse with the order of the characteristic microstructure-length scale. We consider an effective system, which is rigorously obtained via two-scale convergence, and we derive quantitative error estimates.

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Unfolding-based corrector estimates for a reaction–diffusion system predicting concrete corrosion

Cited by 23 publications

References 13 publications

Upscaling of a diffusion problem with interfacial flux jump leading to a modified Barenblatt model

Upscaling of a diffusion problem with interfacial flux jump leading to a modified Barenblatt model

Corrector estimates for the homogenization of a locally periodic medium with areas of low and high diffusivity

Error estimates for nonlinear reaction-diffusion systems involving different diffusion length scales

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