Two multigrid methods for three-dimensional problems with discontinuous and anisotropic coefficients

The subject of this paper is two multigrid methods for the numerical solution of \[ - \nabla \cdot (D(x,y,z)\nabla U(x,y,z)) + \sigma (x,y,z)U(x,y,z) = F(x,y,z) \] in a bounded region $\Omega $ of $R^3 $, where $D = (D^1 ,D^2 ,D^3 ),D^i $ is positive, $i = 1,2,3$, and $D^i ,\sigma $, and F are allowed to be discontinuous across internal boundaries of $\Omega $. The first method differs in two important ways from previously considered multigrid methods for such problems; first, if standard coarsening is used, then anisotropic problems require plane relaxation to obtain a good smoothing factor; for problems with discontinuous coefficients, this was previously done with an ICCG method; here it is done with a two-dimensional multigrid method. Second, a slightly different form of interpolation is used which improves performance for nearly singular problems. Previously considered multigrid methods for problems with discontinuous coefficients have defined the coarse grid operators by Galerkin coarsening; in the second method, such coarsening is also used but with auxiliary intermediate grids obtained by semicoarsening successively in each of the three independent variables; this artifice simplifies coding considerably and results in comparable convergence.