In this paper the optimal consumption strategy of an investor who owns a fixed sized risky project is studied. The cash flows generated by the risky project follow an arithmetic Brownian motion, and the investor earns interest on cash reserves. The short-rate may be stochastic, and the time horizon may be finite. This results in a family of Hamilton-Jacobi-Bellman variational inequalities that include PDEs whose solutions must be approximated numerically. To do so an finite element approximation and a time marching scheme are employed.