  We investigate the wave effects of gravitational waves (GWs) using numerical
simulations with the finite element method (FEM) based on the publicly
available code {\it deal.ii}. We robustly test our code using a point source
monochromatic spherical wave. We examine not only the waveform observed by a
local observer but also the global energy conservation of the waves. We find
that our numerical results agree very well with the analytical predictions.
Based on our code, we study the scattering of GWs by compact objects. Using
monochromatic waves as the input source, we find that if the wavelength of GWs
is much larger than the Schwarzschild radius of the compact object, the
amplitude of the total scattered GWs does not change appreciably due to the
strong diffraction effect, for an observer far away from the scatterer. This
finding is consistent with the results reported in the literature. However, we
also find that, near the scatterer, not only the amplitude of the scattered
waves is very large, comparable to that of the incident waves, but also the
phase of the GWs changes significantly due to the interference between the
scattered and incident waves. As the evolution of the phase of GWs plays a
crucial role in the matched filtering technique in extracting GW signals from
the noisy background, our findings suggest that wave effects should be taken
into account in the data analysis in the future low-frequency GW experiments,
if GWs are scattered by nearby compact objects in our local environment.
