We present a mixed (Chebyshev) collocation-(Fourier) Galerkin projection (MCGP) algorithm that can be used for the computation of the eigenspectra of viscoelastic flows in non-viscometric geometries with a periodic (streamwise) direction. The MCGP algorithm is used to investigate the structure of the eigenspectrum of the viscoelastic periodic channel flow in the purely elastic regime. The numerical eigenspectrum consists of three sets of eigenvalues: (i) poorly resolved continuous set of eigenvalues with real parts close to the -1/We line corresponding to the integration of the viscoelastic stress subject to steady state kinematics; (ii) discrete, mesh-converged eigenvalues that represent the physical stability characteristics of the flow; and (iii) spurious mesh-dependent eigenvalues that are artifacts of the spatial discretization of the differential eigenvalue problem. Since the maximum streamwise wavenumber increases with streamwise refinement, the loss of convergence to the -1/We line of the continuous set of eigenvalues amplifies with increasing mesh refinement. Since the eigenfunctions are singular, wall-normal mesh refinement is ineffective in improving the numerical convergence even for moderate values of the streamwise wavenumber. In addition, the spurious eigenvalues could become more positive with mesh refinement. For larger values of We, a spurious eigenvalue could be more positive than the least negative discrete mesh-converged, physically relevant eigenvalue. Increasing the channel wall amplitude aggravates these problems. These issues present formidable challenges to the development of algorithms for time-dependent viscoelastic flow simulations. Similarly, isolation of the physically relevant eigenvalues from the spurious and the continuous modes using subspace iteration methods could also be difficult. We show that systematic streamwise and wall-normal mesh refinement is required to isolate the physically relevant eigenvalues from the spectrum.