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Steady state,two-dimensional flows may become unstable under two and three-dimensional disturbances, if the flow parameters exceed some critical values.

In many practical situations, determining the parameters at which the flow becomes unstable is essential.

The complete understanding of viscous and viscoelastic flows requires not only the steady state solution of the governing equations, but also its sensitivity to small perturbations.

Linear stability analysis leads to a generalized eigenvalue problem, GEVP.

Solving the GEVP is challenging, even for Newtonian liquids, because the incompressibility constraint creates singularities that lead to nonphysical eigenvalues at infinity.

For viscoelastic flows, the difficulties are even higher because of the continuous spectrum of eigenmodes associated with differential constitutive equations.

The complexity and high computational cost of solving the GEVP have probably discouraged the use of linear stability analysis of incompressible flows as a general engineering tool for design and optimization.

The Couette flow of UCM liquids has been used as a classical problem to address some of the important issues related to stability analysis of viscoelastic flows.

The spectrum consists of two discrete eigenvalues and a continuous segment of eigenvalues with real part equal to -1/We (We is the Weissenberg number).

Most of the numerical approximation of the spectrum of viscoelastic Couette flow presented in the literature were obtained using spectral expansions.

The eigenvalues close to the continuous part of the spectrum show very slow convergence.

In this work, the linear stability of Couette flow of a Newtonian and UCM liquids were studied using finite element method, which makes it easier to extend the analysis to complex flows.

A new procedure to eliminate the eigenvalues at infinity from the GEVP that come from differential equations is also proposed.

The procedure takes advantage of the structure of the matrices involved and avoids the computational effort of common mapping techniques.

With the proposed procedure, the GEVP is transformed into a smaller simple EVP, making the computations more effcient.

Reducing the computational memory and time.

The relation between the eigenvector from the original problem and the reduced one is also presented.