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Singularly perturbed diffusion-advection-reaction processes on extremely large three-dimensional curvilinear networks with a periodic microstructure -- efficient solution strategies based on homogenization theory

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  • Boundary value problems on large periodic networks arise in many applications such as soil mechanics in geophysics or the analysis of photonic crystals in nanotechnology. As a model example, singularly perturbed elliptic differential equations of second order are addressed. Typically, the length of periodicity is very small compared to the size of the covered region. The overall complexity of the networks raises serious problems on the computational side. The high density of the graph, the huge number of edges and vertices and highly oscillating coefficients necessitate solution schemes, where even a numerical approximation is no longer feasible. Realizing that such a system depends on two spatial scales - global scale (full domain) and local scale (microstructure) - a two-scale asymptotic analysis for network differential equations is applied. The limit process leads to a homogenized model on the full domain. The homogenized coefficients cover the micro-oscillations and the topology of the periodic network and characterize the effective behaviour. The approximate model's quality is guaranteed by error estimates. Furthermore, singularly perturbed microscopic models with a decreasing diffusion part and transport-dominant problems are discussed. The effectiveness of the two-scale limit analysis is demonstrated by numerical examples of diffusion-advection-reaction problems on large periodic grids.
    Mathematics Subject Classification: Primary: 4B45, 34E13, 34E05, 34E10.

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