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The stability of a three-dimensional, incompressible, viscous flow through afinite-length duct is studied. A divergence-free basis technique is used toformulate the weak form of the problem. A SUPG (streamingline upwindPetrov-Galerkin) based scheme for eigenvalue problems is proposed to stabilizethe solution. With proper boundary condtions, the least-stable eigenmodes anddecay rates are computed. It is again found that the flows are asymptoticallystable for all $Re$ up to $2500$. It is discovered that the least-stableeigenmodes have a boundary-layer-structure at high $Re$, although thePoiseuille base flow does not exhibits such structure. At these Reynoldsnumbers, the eigenmodes are dominant in the vicinity of the duct wall and areconvected downstream. The boundary-layer-structure brings singularity to themodes at high $Re$ with unbounded perturbation gradient. It is shown that dueto the singular structure of the least-stable eigenmodes, the linearNavier-Stoker operator tends to have pseudospectrua and the nonlinear mechanismkicks in when the perturbation energy is still small at high $Re$. Thedecreasing stable region as $Re$ increases is a result of both the decreasingdecay rate and the singular structure of the least-stable modes. The resultdemonstrated that at very high $Re$, linearization of Navier-Stokes equationfor duct flow may not be a good model problem with physical disturbances.

On the three-dimensional stability of Poiseuille flow in a finite-length duct