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Multiscale analysis and simulation of a reaction-diffusion problem with transmission conditions. (English) Zbl 1228.35030

The multiscale method as a form of homogenization technique is used by the authors for finding solutions to reaction-diffusion problems with some transmission conditions. The paper is well organized and the appropriate references acknowledged. The variational formulation of the problem is stated nicely and is is shown that the sequence \((u^+_{0N}, u^-_{0N}, u_N^M)\) is bounded in the appropriate spaces, enabling them to extract subsequences which converge weakly in the chosen spaces. This enables the authors to obtain the unique weak solution and hence the convergence of the whole sequence of Galerkin approximates. They finally illustrate their results by means of numerical simulations.

MSC:

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35K57 Reaction-diffusion equations
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References:

[1] Neuss-Radu, M.; Jäger, W., Effective transmission conditions for reaction-diffusion processes in domains separated by an interface, SIAM J. Math. Anal., 39, 687-720 (2007) · Zbl 1145.35017
[2] Hoang, V. H.; Schwab, C., High-dimensional finite elements for elliptic problems with multiple scales, Multiscale Model. Simul., 3, 168-194 (2004/05), (electronic) · Zbl 1074.65135
[3] Lions, J. L., Quelques méthodes de résolution des problèmes aux limites non linéaires (1969), Dunod: Dunod Paris · Zbl 0189.40603
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