The multiscale mixed finite-element method is very flexible with respect the choice of both fine-scale and coarse-scale grids. Given a fine-grid solver, basis functions can be defined for almost any collection of connected fine grid cells.

Figures 1 and 2 show two multiscale solutions computed on a unstructured triangular fine grid in two spatial dimensions. The coarse grid blocks are formed as collections of fine-grid triangular cells and can thus be of an almost arbitrary polygonal shape. In Figure 1, the coarse grid is chosen to be structured, meaning that all coarse blocks are (almost) parallelepipeds. In Figure 2, the coarse grid is take to be unstructured and contains cells that are (almost) triangular, quadrilateral, and pentagonal.

Using unstructured triangular grids for the fine-grid model, it is easy to adapt both the fine-grid and the coarse grid to complex external and internal boundaries. Combined with the multiscale methodology, this gives a very flexible approach for performing simulations on high-resolution grid model with complex geometries. We are currently working on extending our prototype code to three spatial dimensions.

Fig. 1: Multiscale mixed finite elements on a structured coarse grid imposed on an unstructured triangular fine grid.

Fig. 2: Multiscale mixed finite elements on a unstructured triangular coarse grid imposed on a unstructured triangular fine grid.