Homogenization of Nonlocal Electrostatic Problems by Means of the Two - Scale Fourier

Multiple scales phenomena are ubiquitous, ranging from mechanical properties of wood, turbulent flow in gases and fluids, combustion, remote sensing of earth to wave propagation or heat conduction in composite materials. The obstacle with multi-scale problems is that they, due to limited primary memory even in the largest computational clusters, can not easily be modeled in standard numerical algorithms. Usually we are not even interested in the fine scale information in the processes. However, the fine scale properties are important for the macroscopic, effective, properties of for example a fiber composite. Attempts to find effective properties of composites dates back more than hundred years, e.g. see Faraday (1965); Maxwell (1954a;b); Rayleigh (1892). One way to find effective properties is to introduce a fine scale parameter, ε > 0, in the corresponding governing equations (modeling fast oscillating coefficients) and then study the asymptotic behavior of the sequence of solutions, and equations, when the fine scale parameter tends to zero. The limit yields the homogenized equations, that have constant coefficients (corresponding to homogeneous material properties). The discipline of partial differential equations dealing with such issues is called homogenization theory. The foundation of homogenization theory was started by Spagnolo (1967) who introduced G-convergence, followed by Γ-convergence by Dal Maso (1993); De Giorgi (1975); De Giorgi & Franzoni (1975); De Giorgi & Spagnolo (1973), and H-convergence Tartar (1977). The two-scale convergence concept introduced by Nguetseng (1989) and developed by Allaire (1992); Allaire & Briane (1996) simplified many proofs. Floquet-Bloch expansion Bloch (1928); Floquet (1883) provides a method to find dispersion relations in the case the fine scales are on the same order as for example the wavelength of a propagating wave. The technique of Floquet-Bloch expansion can also be used to find the classical homogenized properties Allaire & Conca (1996); Bensoussan et al. (1978); Conca et al. (2002); Conca & Vanninathan (1997; 2002). Two-scale transforms have been introduced in different settings, Arbogast et al. (1990); Brouder & Rossano (2002); Cioranescu et al. (2002); Griso (2002); Laptev (2005); Nechvátal (2004). The general idea with the two-scale transform is to map bounded sequences of functions defined on L2(Ω) to sequences defined on the product space L2(Ω × Tn) and then taking the weak limit in L2(Ω× Tn). Besides finding the effective material properties, one can also establish easily computed bounds of these. The boundsmay be as simple as the arithmetic and harmonic averages, or more complex. For further reading we recommend the monograph by Milton (2002) as an introduction to the theory of composites. Homogenization of Nonlocal Electrostatic Problems by Means of the Two-Scale Fourier Transform 10