Abstract The finite-element method is used to solve mixed-, farced-, and natural-convection problems in two-dimensional incompressible laminar flows. The streamfunction-vorticity equations are uncoupled and solved in sequence with the energy equation. The wall vorticity is evaluated in the framework of the streamfunction equation, and particular care is taken to specify inflow and outflow boundary conditions properly. The resulting scheme achieves convergence without the traditional need for upwinding, even for very high values of the Reynolds and the Rayleigh numbers. Stability and accuracy of the approach are demonstrated by the solution of three well-known test problems concerning mixed and forced convection downstream of a backward-facing step, and natural convection in a heated square cavity.