  This work establishes the weak convergence of Euler-Maruyama's approximation
for stochastic differential equations (SDEs) with singular drifts under the
integrability condition in lieu of the widely used growth condition. This
method is based on a skillful application of the dimension-free Harnack
inequality. Moreover, when the drifts satisfy certain regularity conditions,
the convergence rate is estimated. This method is also applicable when the
diffusion coefficients are degenerate. A stochastic damping Hamiltonian system
is studied as an illustrative example.
