Multi-scale methods

Multi-scale methods are appropriate when detailed information about the microscopic structure of the solution (i.e. the structure of the solution at scale 'eps') are needed. This may occur for instance when nonlinear interactions at the microscopic level generate macroscopic structures which persist in the limit eps -> 0 .

To perform simulations at the macro-scale level, some knowledge of the local micro-scale structure is needed. Multiscale methods aim at providing this information thanks to sub-grid calculations (see schematics of the picture above: the left picture shows the macroscopic grid. The right picture provides a blow-up of one of the macroscopic grids to highlight the subgrid calculation. The subgrid calculation is purely local). The macro-scale solution is then sought as some combination of these local sub-grid solutions. This idea gives rise to many variants, Here, we present:

• • The Macro-Guided Micro (MGM) method

  • The Localized Model Upscaling (LMU) method

  • The MS-FEM method for perforated domains.

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