MsFEM

MsFEM, the Multiscale Finite Element Method, is a numerical approach for solving partial differential equations with highly heterogeneous or multiscale coefficients. It combines the finite element framework with basis functions that encode fine-scale information, enabling accurate simulations on a coarse computational grid.

The method constructs a coarse mesh covering the domain. For each coarse element, local boundary-value problems are solved to generate multiscale basis functions that reflect the underlying microstructure. These basis functions are then assembled into a global coarse-scale system, whose solution provides the macroscopic behavior, from which the fine-scale details can be recovered through the basis expansion. MsFEM is particularly effective for elliptic and parabolic problems with rapidly oscillating or high-contrast material properties, where standard finite elements require prohibitively fine grids.

Several variants and extensions exist. Oversampling techniques reduce boundary-induced errors in the local problems. Discontinuous Galerkin formulations, and combinations with variational multiscale methods, improve stability and flexibility, especially for complex geometries or nonconforming meshes. MsFEM has adaptations for time-dependent problems and nonlinear systems, often via linearization or iterative schemes. It can also be integrated with stochastic or uncertain data frameworks.

Common applications include flow in porous media and reservoir simulation, elasticity and composite materials, groundwater transport, and geophysical problems. Strengths of MsFEM lie in efficiently capturing fine-scale effects on a coarse grid and handling high-contrast multiscale coefficients. Limitations include potential resonance errors with certain boundary conditions, the need for carefully designed local problems, and computational cost associated with precomputing multiscale basis functions.