A novel control-volume finite-element method for heat transfer and fluid flow, in which all variables are colocated, is demonstrated and validated by application to a variety of test problems. The test problems are the convective scalar transport of a step profile at various angles to the grid and the fluid flow prediction for the driven cavity problem with Reynolds numbers of 100, 400, and 1000, for inviscid flow across a forward-facing step, and for rigid-body rotation discretized on a rectangular mesh with Reynolds numbers of 28.3 and infinity. The performance of the method is found to be excellent in terms of accuracy, providing excellent agreement with exact and grid-independent results, using only modest mesh discretization. The performance of the rigid-body rotation problem indicates that the scheme is “almost” second-order accurate.