In this paper, a finite element numerical approach to the solution of linear stability problems is described. The perturbation equation governing the linear instability waves in confined supersonic jets is evaluated using a global scheme instead of the traditional shooting method. Typical global schemes such as the finite difference method and the boundary element method have been applied previously to solve related problems for circular and elliptic jets. To improve the robustness of the calculation for more complex jet exit geometries, the present study makes use of the finite element method (FEM). Additional accuracy for the calculations is achieved through grid refinement by re-generating the FEM grid manually. A local iterative method is then used to obtain refined eigensolutions. Favorable agreement between the FEM and semi-analytical results is achieved. The effects of variations of the shroud geometry on the instability wave characteristics of confined circular and elliptic jets are also analyzed. In the case of a circular jet in a non-circular duct, a decrease in the growth rate of the Kelvin-Helmholtz instability wave is observed. It is proposed that such a decrease in growth rate is due to the decrease in the perturbation pressure gradient inside the mixing layer. For confined elliptic jets, the effects of confinement on the different classes of instability modes are documented. The mode that ''flaps'' about the jet major axis is found to be less influenced by the presence of a shroud than the ''varicose'' mode that is even about both the jet major and minor axes.
