Multiscale modeling of materials requires simulations
of multiple levels of structural hierarchy. The computational efficiency
of numerical methods becomes a critical factor for simulating
large physical systems with highly desperate length scales.
A flow with small particles represents a particularly difficult problem for
efficient multigrid solution. These particles are often so small,
occupying just a few points
on the fine grid, that they cannot be properly incorporated
into a coarse-grid formulation. Efficiency, i.e., convergence rate,
of classical multigrid solvers
deteriorates significantly because of the poor accuracy of coarse-grid
In this talk, I am planning to overview and evaluate several existing multigrid approaches to solving problems with small particles. A new multigrid method based on a modified local Galerkin coarsening scheme will be presented. Numerical tests confirm recovering the optimal (Laplace-like) efficiency in solution of practically interesting problems.