​​We present a discontinuous Galerkin method for site effects assessment. The P–SV seismic wave propagation is studied in 2-D space heterogeneous media. The first-order velocity–stress system is obtained by assuming that the medium is linear, isotropic and viscoelastic, thus considering intrinsic attenuation. The associated stress–strain relation in the time domain being a convolution, which is numerically intractable, we consider the rheology of a generalized Maxwell body replacing the convolution by a set of differential equations. This results in a velocity–stress system which contains additional equations for the anelastic functions expressing the strain history of the material. Our numerical method, suitable for complex triangular unstructured meshes, is based on centred numerical fluxes and a leap-frog time-discretization. The method is validated through numerical simulations including comparisons with a finite-difference scheme. We study the influence of the geological structures of the Nice basin on the surface ground motion through the comparison of 1-D and 2-D soil response in homogeneous and heterogeneous soil. At last, we compare numerical results with real recordings data. The computed multiple-sediment basin response allows to reproduce the shape of the recorded amplification in the basin. This highlights the importance of knowing the lithological structures of a basin, layers properties and interface geometry.