We discuss theoretical and computational method on plate-driven mantle flow beneath mid-ocean ridges. We consider a steady-state flow induced by motion of overlying rigid plates in an incompressible viscous mantle beneath a generic ridge-transform-ridge plate boundary. No assumption of orthogonal and symmetric spreading at ridge axis is made. Analytical solutions for viscosity flow in a half-space and in a layered viscosity mantle beneath an infinitesimal thickness lithosphere and beneath plates that thicken with increasing age, are presented. Numerical calculations were carried out using a standard fast Fourier transform algorithm. The difficulty of using standard Fourier methods to predict accurately the mantle flow field in the proximity of the plate boundaries is overcome by applying the Gegenbauer reconstruction post-processing technique to the Fourier pseudo-spectral solutions. Finally, we present some examples of flow computations. We consider, for both models, two different ridge-transform-ridge geometries consisting of 100 and 1000 km offsets of two ridge segments spreading at 15 mm/yr half rate. We found a significant difference in the flow structure between the two flow models close to ridge axis and ridge-transform intersections. The proposed model and methods are useful for fast mantle flow calculations to investigate melting processes beneath spreading centres, and to predict the relationship between mantle temperature, crustal thickness and geochemistry of the oceanic crust.