A high-order corrector estimate for a semi-linear elliptic system in perforated domains

Khoa, Vo Anh

doi:10.1016/j.crme.2017.03.003

Cited by 3 publications

(5 citation statements)

References 15 publications

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“…Due to the structure of the auxiliary problems (30)-(36), we get u k,l ≡ 0 for all k ≥ 1 and (k, l) ∈ K α,θ . In line with [15], we obtain when k = 0 that…”

Section: Due To the Simple Relationsupporting

confidence: 75%

“…The use of this cut-off function to prove the convergence rates is not only seen in elliptic problems that we have taken into consideration, but also can be found in some particular multiscale models. Aside from our earlier works [15,16], this technique is applied in the works [34,19] for a nonlinear drift-reaction-diffusion model in a heterogeneous solid-electrolyte composite and in [36] in the context of phase field equations. Besides, we single out the survey [43] and the work [40] for a concrete background of the so-called operator corrector estimates related to this approach.…”

Section: (44)mentioning

confidence: 99%

“…The high-order corrector estimates we prove in this paper are involving the presence of the scaling parameters. This is the extended follow-up result of our earlier works [16,17,15], where we wish to estimate the differences between micro-macro concentrations and micro-macro concentration gradients in the energy space of perforated domains. It is also in the same line with the theoretical findings in [20,5,3,2,27,28,12].…”

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confidence: 92%

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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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“…Due to the structure of the auxiliary problems (30)-(36), we get u k,l ≡ 0 for all k ≥ 1 and (k, l) ∈ K α,θ . In line with [15], we obtain when k = 0 that…”

Section: Due To the Simple Relationsupporting

confidence: 75%

Section: (44)mentioning

confidence: 99%

mentioning

confidence: 92%

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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 follow up on our earlier works (Khoa and Muntean 2016;Khoa 2017;Khoa et al 2020) that focus on the asymptotic analysis of semi-linear elliptic problems posed in perforated domains. This well-understood elliptic problem was applied in the studies of the heat transfer in composite materials and of the pressure and phase velocities in porous media flow.…”

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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“…In this paper, we follow up on our earlier works [1,2] that focus on the asymptotic analysis of semi-linear elliptic problems posed in perforated domains. Cf.…”

Section: Introductionmentioning

confidence: 99%

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

Vo¹,

Ijioma²,

Nguyen³

2019

Preprint

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This work is devoted to the development and analysis of a linearization algorithm that can be used for the numerical implementation of 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 L 2 norm for both the microscopic and macroscopic problems. In the standard variational setting, we prove convergence 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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A high-order corrector estimate for a semi-linear elliptic system in perforated domains

Cited by 3 publications

References 15 publications

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

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

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

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