Homogenization of low-cost control problems on perforated domains

Muthukumar, T.; Nandakumaran, A. K.

doi:10.1016/j.jmaa.2008.09.048

Cited by 14 publications

(12 citation statements)

References 13 publications

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“…e.g. Douanla and Svanstedt 2012; Gryshchuk and de Cristoforis 2013), while it also concerns low-cost control problems in Muthukumar and Nandakumaran (2009). On top of that, they can be further adapted to complex scenarios that include, for example, the non-stationary Stokes-Nernst-Planck-Poisson system in the dynamics of colloids (cf.…”

Section: Background and Motivationmentioning

confidence: 99%

Strong convergence of a linearization method for semi-linear elliptic equations with variable scaled production

Khoa

Ijioma²,

Ngoc

2020

Comp. Appl. Math.

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This work is devoted to the development and analysis of a linearization algorithm for microscopic elliptic equations, with scaled degenerate production, posed in a perforated medium and constrained by the homogeneous Neumann-Dirichlet boundary conditions. This technique plays two roles: to guarantee the unique weak solvability of the microscopic problem and to provide a fine approximation in the macroscopic setting. The scheme systematically relies on the choice of a stabilization parameter in such a way as to guarantee the strong convergence in H 1 norm for both the microscopic and macroscopic problems. In the standard variational setting, we prove the H 1-type contraction at the micro-scale based on the energy method. Meanwhile, we adopt the classical homogenization result in line with corrector estimate to show the convergence of the scheme at the macro-scale. In the numerical section, we use the standard finite element method to assess the efficiency and convergence of our proposed algorithm.

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Section: Background and Motivationmentioning

confidence: 99%

Strong convergence of a linearization method for semi-linear elliptic equations with variable scaled production

Khoa

Ijioma²,

Ngoc

2020

Comp. Appl. Math.

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“…Remark 3. In the sequel, our new assumptions (24) and (25) on the reaction rate R are termed as (A 5 ) and (A 6 ), respectively. It resembles the definition of almost additive functions with positive homogeneity in stochastic processes (see, e.g., [38]).…”

Section: 1mentioning

confidence: 99%

“…As to the literature on this topic, the reader can be referred to [32,31,19,8,33], where the variable scaling has been considered earlier in complex diffusion problems of charged colloidal particles. Besides, the case α > 0 can be related to the context of low-cost control problems on perforated domains in [25].…”

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confidence: 99%

On a pore-scale stationary diffusion equation: Scaling effects and correctors for the homogenization limit

Khoa¹,

Thieu²,

Ijioma³

2021

DCDS-B

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In this paper, we consider a microscopic semilinear elliptic equation posed in periodically perforated domains and associated with the Fouriertype condition on internal micro-surfaces. The first contribution of this work is the construction of a reliable linearization scheme that allows us, by a suitable choice of scaling arguments and stabilization constants, to prove the weak solvability of the microscopic model. Asymptotic behaviors of the microscopic solution with respect to the microscale parameter are thoroughly investigated in the second theme, based upon several cases of scaling. In particular, the variable scaling illuminates the trivial and non-trivial limits at the macroscale, confirmed by certain rates of convergence. Relying on classical results for homogenization of multiscale elliptic problems, we design a modified two-scale asymptotic expansion to derive the corresponding macroscopic equation, when the scaling choices are compatible. Moreover, we prove the high-order corrector estimates for the homogenization limit in the energy space H 1 , using a large amount of energy-like estimates. A numerical example is provided to corroborate the asymptotic analysis.

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“…Different homogenization results for stationary problems in ε-periodic perforated domains have been studied in [4,29,37]. For previous homogenization results in the case of weakly converging data, we quote Tartar (see [9], Proposition 8.17, Remark 8.18 and Theorem 8.19) and [13,52]. As regards evolution problems in domains with imperfect interface, we refer to [21,22,23,54,55].…”

Section: Introductionmentioning

confidence: 99%

Homogenization and exact controllability for problems with imperfect interface

Monsurrò¹,

Perugia²

2019

Networks &Amp; Heterogeneous Media

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In this paper we study the internal exact controllability for a second order linear evolution equation defined in a two-component domain. On the interface we prescribe a jump of the solution proportional to the conormal derivatives, meanwhile a homogeneous Dirichlet condition is imposed on the exterior boundary. Due to the geometry of the domain, we apply controls through two regions which are neighborhoods of a part of the external boundary and of the whole interface, respectively. Our approach to internal exact controllability consists in proving an observability inequality by using the Lagrange multipliers method. Eventually we apply the Hilbert Uniqueness Method, introduced by J. -L. Lions, which leads to the construction of the exact control through the solution of an adjoint problem. Finally we find a lower bound for the control time depending not only on the geometry of our domain and on the matrix of coefficients of our problem but also on the coefficient of proportionality of the jump with respect to the conormal derivatives.

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Homogenization of low-cost control problems on perforated domains

Cited by 14 publications

References 13 publications

Strong convergence of a linearization method for semi-linear elliptic equations with variable scaled production

Strong convergence of a linearization method for semi-linear elliptic equations with variable scaled production

On a pore-scale stationary diffusion equation: Scaling effects and correctors for the homogenization limit

Homogenization and exact controllability for problems with imperfect interface

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