<p>This work is about a new two-level solver for Helmholtz equations discretized by finite elements. The method is inspired by two-grid methods for finite-difference Helmholtz problems as well as by previous work on two-level domain-decomposition methods. For the coarse-level discretization, a compact-stencil finite-difference method is used that minimizes dispersion errors. The smoother involves a domain-decomposition solver applied to a complex-shifted Helmholtz operator. Local Fourier analysis shows the method is convergent if the number of degrees of freedom per wavelength is larger than some lower bound that depends on the order, e.g., more than 8 for order 4. In numerical tests, with problem sizes up to 80 wavelenghts, convergence was fast, and almost independent of problem size unlike what is observed for conventional methods. Analysis and comparison with dispersion-error data shows that, for good convergence of a two-grid method for Helmholtz problems, it is essential that fine- and coarse-level dispersion relations closely match.</p>