An efficient finite-element approach using 1D Hermitian polynomials is proposed for dispersion and modal analysis of weakly guiding fibers with arbitrary refractive-index profile. Hermitian line elements consider both the field and its derivative at the nodal point, therefore, the continuity of the field and its derivative between elements can be fully satisfied. This feature highly enhance the accuracy of the finite-element solution. The present approach is validated through the analysis of several profiles, including dispersion-flattened fibers.