  Phonons are quantized vibrations of a crystal lattice that play a crucial
role in understanding many properties of solids. Density functional theory
(DFT) provides a state-of-the-art computational approach to lattice vibrations
from first-principles. We present a successful software implementation for
calculating phonons in the harmonic approximation, employing density-functional
perturbation theory (DFPT) within the framework of the full-potential
linearized augmented plane-wave (FLAPW) method as implemented in the electronic
structure package FLEUR. The implementation, which involves the Sternheimer
equation for the linear response of the wave function, charge density, and
potential with respect to infinitesimal atomic displacements, as well as the
setup of the dynamical matrix, is presented and the specifics due to the
muffin-tin sphere centered LAPW basis-set and the all-electron nature are
discussed. As a test, we calculate the phonon dispersion of several solids
including an insulator, a semiconductor as well as several metals. The latter
are comprised of magnetic, simple, and transition metals. The results are
validated on the basis of phonon dispersions calculated using the finite
displacement approach in conjunction with the FLEUR code and the phonopy
package, as well as by some experimental results. An excellent agreement is
obtained.
