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 approximations.
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.