Geometrically stiff problems - problems which exhibit disparate length scales geometric in origin - arise frequently in fluid mechanics and heat transfer, perhaps most prevalently in the analysis of multicomponent media. In this paper we present a hybrid analytico-computational approach to the treatment of geometrically stiff problems for which the quantity of interest can be expressed as the extremum of a functional; more precisely, we consider, in depth, the effective conductivity of (random) thermal composites and, briefly, the permeability of (random) porous media. The approach proceeds by an inner-outer decomposition, in which analytical approximations in 'inner' nip regions are folded into modified 'outer' problems defined over geometrically (more) homogeneous domains. The technique considerably mitigates the numerical difficulties associated with geometric stiffness - by reduced degrees of freedom, improved conditioning, less stringent mesh generation requirements and better parallel load balancing - while simultaneously providing rigorous upper and lower bounds for the quantity of interest. Furthermore, the method enjoys certain 'convergence' properties: the upper-lower bound gap can be rendered systematically smaller by decreasing the extent of the inner nip regions. We first describe a series of bound procedures and associated modified outer weak forms; we next discuss finite-element implementation of the outer problem on message-passing multicomputers (here the Intel iPSC/860 hypercube); and, finally, we present several illustrative examples.
