The flow patterns in a steady, viscous L-shaped cavity are investigated using both analytic solutions combined with methods from nonlinear dynamical systems and numerically using the finite element method. A boundary value problem is formulated for the case of Stokes flow, which is solved analytically, , expressed as an infinite series of eigenfunctions which depend on and which are related to the heights of the L-shaped domain. The control space diagram is constructed for exhibiting the mechanism by which new eddies are obtained in the cavity.