Phase Diagrams of Hubbard Models with Small Unit Cells

We present a controlled approach to the low temperature phase diagram of highly inhomogeneous Hubbard models in the limit of small coupling, $t'$, between clusters. We apply this to the dimerized and checkerboard models with any strength of $U$. The dimerized model is found to behave like a doped semiconductor, with a Fermi-liquid groundstate with parameters ({\it e.g.} the effective mass) which are smooth, and unspectacular functions of $U$. By contrast, the checkerboard model has a Fermi liquid phase at large $U > U_c = 4.67$, a d-wave superconducting state with a full gap for $U_c > U > 0$, and a narrow strip of an intermediate d-wave superconducting phase with gapless ``nodal'' quasiparticles for $|U - U_c| < {\cal O}(t^\prime)$.