We present a novel full-waveform inversion approach which can reduce the computational cost by up to an order of magnitude compared to conventional approaches, provided that variations in medium properties are sufficiently smooth. Our method is based on the usage of wavefield-adapted meshes which accelerate the forward and adjoint wavefield simulations. By adapting the mesh to the expected complexity and smoothness of the wavefield, the number of elements needed to discretise the wave equation can be greatly reduced. This leads to spectral-element meshes which are optimally tailored to source locations and medium complexity. We demonstrate a workflow which opens up the possibility to use these meshes in full-waveform inversion and show the computational advantages of the approach. We provide examples in 2-D and 3-D to illustrate the concept, describe how the new workflow deviates from the standard full-waveform inversion workflow, and explain the additional steps in detail.