Black Box Multigrid for Checkerboard Problems
In the 1981 paper by Alcouffe, et. al., the two-dimensional diffusion equation with discontinuous diffusion coefficients is considered. Finite volume discretization is considered, giving rise to a five-point operator on the finest grid. Example II in that paper is a problem with a four corner junction problem, otherwise known as a checkerboard problem. The diffusion coefficient is constant on four quadrants that meet at the center point, with values in the northwest, northeast, southwest, and southeast quadrants of 1000, 1, 1, and 1000, leading to values of 500.5 at the center point. Results are given for these values and for 500.5 replaced by, presumably, 1. Reference is made to modification using (5.8), but the discussion in that section of the paper is in terms of normalized coefficients 1, &epsi#epsilon;, &epsi#epsilon;, and 1. Therefore, it is not clear what problem is being reported on, what the convergence criterion is, etc. In any case, Alan Chen of the University of Washingtion was unable to obtain good convergence for this modified problem using the current version of BoxMG, with red-black point relaxation. Curiously, convergence is good for the values 500.5. Brandt's motivation for replacing 500.5 by, presumably 1, is to facilitate communication between the northeast and southwest regions. We show how to modify BoxMG to achieve good convergence for this modified problem. The modification is consistent with Brandt's intuition. We also investigate the adaptive method of Chartier, MacLachlan, and Moulton for the modified problem. Finally, AMG would presumably perform well for the modified problem, since the weak connections to the center point would lead to the center point not being a grid point on coarser grids. Therefore, there may be implications for improving AMG in the cases that bad coarse grids are chosen.