Contour plot of the y -component of the adjoint velocity u∗and of the adjoint potential Φ∗. (a) y-component of adjoint velocity u∗. It is mainly different from zero inside the vertical channel indicating that the primal solution needs to be accurate in that region. (b) Adjoint potential Φ∗. Note that Φ∗ almost vanishes everywhere except at the corners and along the middle section of the top wall.