A new numerical technique for solving the 2D elastodynamic equations in the frequency domain based on a finite-volume P0 approach is proposed for application to full waveform inversion. The associated discretisation is through triangles and the free surface is described along the edges of the triangles which may have different slopes. By applying a parsimonious strategy, only velocity fields are left as unknowns in triangles, minimizing the core memory requirement of the simulation. Efficient PML absorbing conditions have been designed for damping waves around the grid. The method is validated against analytical solutions of several canonical problems and with numerical solutions computed with a well-established finite-difference time-domain method in heterogeneous media. In presence of a free surface, the finite-volume method requires ten triangles per wavelength for a flat topography and fifteen triangles per wavelength for more complex shapes, well below criteria required by the staircase approximation of finite-difference methods. Comparison between the frequency-domain finite-volume and the second-order rotated finite-difference methods also shows that the former is faster and less-memory demanding for a given accuracy level, an encouraging point for application of full waveform inversion in realistic configurations. ...