Since the late 1950s, alternating direction implicit (ADI) methods have been used for modeling diffusive heat transfer and analogous physical processes. In most cases, ADI has been applied to problems with uniform orthogonal grids (UOGs), due to directional operator splitting. Based on UOGs, 2D Peaceman-Rachford ADI and 3D Douglas-Brian ADI are second order accurate in space (and time when modeling time-dependent linear diffusion). With appropriate time-step adaption, these implementations require as few as O(logN) iterations to converge to steady state solutions over N unknowns. They also exhibit good parallel scalability down to fairly small numbers of unknowns, using spatial domain decomposition. But since UOGs are incompatible with most ways of resolving multiple spatial scales, ADI has fallen to the wayside of less spatially structured discretizations on top of Krylov and multigrid iterative linear solvers. However, it should be realized that UOGs are only a subset of the grids on which ADI can be defined. ADI discretizations and iterations can be defined for locally refined orthogonal composite grids (LROCGs) as well. This could significantly benefit many LROCG block-structured AMR simulations of diffusive processes. Conceptually, local refinement merely increases the complexity of the line- solves in ADI. In particular, a graph showing interdependence of unknowns along a particular direction is no longer simply-connected. Such graphs take on topologies resembling vascular capillaries. For one of the simplest conservative spatial discretizations, traversal of a coarse-fine boundary into the finer region links the coarser unknown by matrix coefficients to r^(d-1) finer unknowns, assuming the same refinement ratio (r) for the (d) dimensions of space. However, the finer unknowns at the boundary are not linked by matrix coefficients to one another. The spatial accuracy for such a coarse-fine discretization is reduced to first order, but the linear system of equations for the capillary requires few more FLOPs than a compact tridiagonal system with the same number of unknowns. In particular, for N "capillary" unknowns and a fixed number of refinement levels, ADI still takes only O(N) FLOPs. However, for efficiency, the solvers require a carefully structured and sufficiently general Gaussian elimination algorithm greatly exceeding the complexity of the compact tridiagonal algorithm. For this reason initial studies of LROCG ADI might use almost any iterative solvers for the capillaries. In particular, the Portable Extensible Toolkit for Scientific Computations (PETSc), is being investigated due to parallelizability. Overall parallelism of a block-structured AMR based on ADI should not be significantly degraded from UOG ADI once direct capillary solvers commensurate with spatial domain decomposition are implemented. These direct capillary solvers might explicitly take advantage of block-wise homogeneous and particular solutions to diminish serial bottlenecks. Whether such direct solvers are actually necessary is to be determined. This presentation should illuminate: time-dependent accuracy and steady-state convergence properties of LROCG ADI, discretization error differences between LROCG and UOG ADI, and preliminary code timings and timing estimates for LROCG ADI.