Darcy Flow, Multigrid, and Upscaling
James M. Rath · Kluwer Academic Publishers eBooks · 2006
Simulation of flow through a heterogeneous porous medium with fine-scale features can be computationally expensive if the flow is fully resolved. Multigrid algorithms are often used to compute approximations to such problems because of storage considerations, the high condition number of the problem, and multigrid’s resolution independent convergence rate. The proposed algorithm takes an ordinary multigrid method and replaces the coarsening procedure with an upscaling one. Upscaling aims for performing computations on a coarser scale but still retaining information about the fine-scale flow and problem data. That is, an upscaled computation is just as cheap as a coarse one, but much more accurate. By replacing coarsening with upscaling in multigrid, faster convergence ensues. Numerous upscaling techniques have been proposed; variational multiscale sub-grid upscaling is used here. By keeping within the usual finite element variational framework, known multigrid analysis techniques can be applied. Like multigrid, the proposed scheme has a convergence factor dependent on both the permeability field (the ratio of maximum to minimum values) and the relative resolution of the fine and coarse grids.