The decomposed element-free Galerkin (DEFG) method is a modified scheme to resolve shortcomings of memory use in element-free Galerkin (EFG) methods. DEFG solves elastic wave equation problems by alternately updating the stress-strain relations and the equations of motion as in the staggered-grid finite-difference (FD) method. DEFG requires at most twice the memory space, a size comparable to that used by the FD method. In addition, DEFG can adopt perfectly matched layer (PML) absorbing boundary conditions as in the FD case. To confirm that DEFG performs as well as FD, a 2D DEFG under PML boundary conditions was compared with an FD with fourth-order spatial accuracy (FD4) using an exact analytical solution of PS reflection waves. The DEFG results are as accurate as those obtained by FD4. In a comparison using Lamb's problem with eight nodal spaces for the shortest S-wavelength, DEFG provides a remarkably accurate Rayleigh waveform over a distance of at least 50 wavelengths compared with 10 wavelengths for FD4. In the Rayleigh-wave case, DEFG with grid spacing is more accurate than FD4 with grid spacing, and DEFG uses less CPU time. DEFG may be a suitable method for numerical simulations of elastic wavefields, especially where a free surface is considered.