We present a finite-difference formulation for 3-D elastic flexure of the lithosphere, which is solved by a direct-matrix method. to incorporate the effect of spatial variations in rigidity, additional terms for the bi-harmonic 3-D flexure equation have been derived from a variational displacement formulation as used in finite-element methods. Additionally, planar faults are treated as discontinuities. These are implemented by an additional degree of freedom for fault heave, and a coupled continuum equation for zero-differential tilting across the fault. the 3-D finite-difference results have been tested for line loads, point loads and disc loads by analytical solutions, and for spatial variation in effective elastic thickness (EET) by 2-D finite-difference solutions. Fault-related flexure patterns are compared to the 2-D analytical broken-plate model developed by Vening-Meinesz (1950). We subsequently apply the 3-D fault model to investigate fault controlled 3-D basement geometries in Lake Tanganyika (East Africa). We show that our model is capable of predicting 3-D basement geometries, characteristically observed in rifted basins. the modelling results indicate that fault-controlled upper crustal flexure patterns are associated with low values for EET. A comparison with regional scale-model studies, showing a superposition of high EET flexure effects, supports a multilayered rheological control on continental rifting.