Multiskalen

Multiskalen, or multiscale modeling, refers to mathematical and computational approaches for systems in which important processes occur at multiple spatial or temporal scales. In such systems, fine-scale phenomena influence coarse-scale behavior, and direct simulation at all scales is often impractical. The aim is to predict macroscopic behavior while either deriving effective equations for large scales or coupling models across scales in a coherent framework.

Key ideas include upscaling, which aggregates fine-scale details into coarse-scale descriptions, and downscaling or downfolding, which preserves essential microscopic information when needed. Common theoretical foundations are provided by homogenization theory, asymptotic analysis, and multiscale expansions. On the numerical side, several methods have become standard: multiscale finite element method (MsFEM); heterogeneous multiscale method (HMM); equation-free approaches that do not assume an explicit macroscopic model; and wavelet-based or other multiresolution techniques for data-driven scale separation. In some settings, concurrent multiscale modeling couples a fine-scale model to a coarser model within a single simulation, while sequential approaches compute effective properties before running larger-scale simulations.

Applications span materials science (composites and porous media), geophysics (reservoir and groundwater flow), biomechanics, and climate or environmental modeling, where processes range from atomic or pore-scale to continuum scales. Multiskalen methods continue to evolve with advances in computation and data-driven modeling, aiming to improve accuracy and efficiency while maintaining physical fidelity.