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

Reichelt, Sina

doi:10.1088/1742-6596/727/1/012013

Cited by 2 publications

(2 citation statements)

References 31 publications

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“…The main purpose of this paper is to obtain corrector estimates that delimitate the error made when homogenizing 10 (averaging, upscaling, coarse graining...) the problem (P ε ), i.e. we want to estimate the speed of convergence as ε → 0 of suitable norms of differences in micro-macro concentrations and micro-macro concentration gradients.…”

Section: Introductionmentioning

confidence: 99%

“…It is worth noting that this work goes along the line open by our works [7] (correctors via periodic unfolding) and [8] (correctors by special test functions adapted to the local periodicity of the microstructures). An alternative strategy to derive correctors for our scenario could in principle exclusively rely on periodic unfolding, refolding and defect operators approach if the boundary conditions along the microstructure would be of homogeneous Neumann type; compare [9] and [10].…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic analysis of a semi-linear elliptic system in perforated domains: Well-posedness and correctors for the homogenization limit

Khoa

Muntean

2016

Journal of Mathematical Analysis and Applications

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In this study, we prove results on the weak solvability and homogenization of a microscopic semi-linear elliptic system posed in perforated media. The model presented here explores the interplay between stationary diffusion and both surface and volume chemical reactions in porous media. Our interest lies in deriving homogenization limits (upscaling) for alike systems and particularly in justifying rigorously the obtained averaged descriptions. Essentially, we prove the well-posedness of the microscopic problem ensuring also the positivity and boundedness of the involved concentrations and then use the structure of the two scale expansions to derive corrector estimates delimitating this way the convergence rate of the asymptotic approximates to the macroscopic limit concentrations. Our techniques include Moser-like iteration techniques, a variational formulation, two-scale asymptotic expansions as well as energy-like estimates. justify rigorously the upscaled models derived in [1] and prepare the playground to obtain corrector estimates for the thermo-diffusion scenario discussed in [5]. From the corrector estimates perspective, the major mathematical difficulty we meet here is the presence of the nonlinear surface reaction term. To quantify its contribution to the corrector terms 15 we use an energy-like approach very much inspired by [6]. The main result of the paper is Theorem 10 where we state the corrector estimate. It is worth noting that this work goes along the line open by our works [7] (correctors via periodic unfolding) and [8] (correctors by special test functions adapted to the local periodicity of the microstructures). An alternative strategy to derive correctors for our scenario could in principle exclusively rely on periodic unfolding, refolding and defect operators approach if the boundary conditions along the microstructure would be of homogeneous Neumann 20 type; compare [9] and [10].The corrector estimates obtained with this framework can be further used to design convergent multiscale finite element methods for the studied PDE system (see e.g. [11] for the basic idea of the MsFEM approach and [12] for an application to perforated media).The paper is organized as follows: In Section 2 we start off with a set of technical preliminaries focusing especially 25 on the working assumptions on the data and the description of the microstructure of the porous medium. The weak solvability of the microscopic model is established in Section 3. The homogenization method is applied in Section 4 to the problem (P ε ). This is the place where we derive the corrector estimates and establish herewith the convergence rate of the homogenization process. A brief discussion (compare Section 5) closes the paper. Preliminaries30 2.1. Description of the geometryThe geometry of our porous medium is sketched in Figure 2.1 (left), together with the choice of perforation (referred here to also as "microstructure") cf. Figure 2.1 (right). We refer the reader to [13] for a concise mathematical representation of the perfo...

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Section: Introductionmentioning

confidence: 99%