Artificial damping in multigrid methods

When the solution and problem coefficients are highly oscillatory, the computed solution may not show characteristics of the original physical problem unless the numerical mesh is sufficiently fine. In the case, the coarse grid problem of a multigrid (MG) algorithm must be still huge and poorly-conditioned, and therefore, it is hard to solve by either a direct method or an iterative scheme. This article suggests a MG algorithm for such problems in which the coarse grid problem is slightly modified by an artificial damping (compressibility) term. It has been numerically observed that the artificial damping, even if slight, makes the coarse grid problem much easier to solve, without deteriorating the overall convergence rate of the MG method. For most problems, 2–6 times speed up have been observed.