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

Abstract

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...

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

“…As main tool, we follow the standard approach by the energy-like method to investigate the error estimate between the micro and macro concentrations and micro and macro concentration gradients. This work aims at generalizing the results reported in [2,7]. …”

“…Essentially, we have analyzed the solvability of the microscopic system in [7], derived the upscaled equations as well as the corresponding effective coefficients, and proved the high-order corrector estimates for the differences of concentrations and their gradients in which the standard energy method has been used. Furthermore, we also solved a reduced similar problem where the Picard iterations-based method is applied to deal with the nonlinear auxiliary problems ( [8]).…”

“…Notice that the quantity ε is called the homogenization parameter or the scale factor. The perforated domain Ω ε ⊂ R d herein approximates a porous medium and its precise description can be found in [5,7]. As an example, we depict in Figure 2.1 an admissible geometry of our medium and the corresponding microstructure.…”

“…In [7], the investigated microscopic semi-linear system resembles a steady state-type of thermo-diffusion systems ( [10,9]). Essentially, we have analyzed the solvability of the microscopic system in [7], derived the upscaled equations as well as the corresponding effective coefficients, and proved the high-order corrector estimates for the differences of concentrations and their gradients in which the standard energy method has been used.…”

“…It is worth noting that the following errors have been so far obtained for the homogenization of the above-mentioned microscopic elliptic system in [7] and further in [2]:…”