Spectral element spatial discretizations are applied to the solution of a one-dimensional continuum model for radio-frequency glow discharges. Time-periodic solutions are computed for the resulting set of differential-algebraic equations using a Newton shooting algorithm that gives quadratic convergence to the solution for a good initial guess. The accuracy of the calculations is shown to exhibit the expected exponential convergence with increasing polynomial order of the Legendre polynomials in each spectral element. Comparison with previously reported calculations using upwind finite difference discretization of the same continuum model shows the smoothing of peaks in the solution structure by the upwind approximation, which may cause up to 20% error in the average energy per electron. Differences between applying a sinusoidal current and a sinusoidal voltage to sustain the plasma are discussed.