  We study approximations of reflected It\^o diffusions on convex subsets $D$
of $\Rd$ by solutions of stochastic differential equations with penalization
terms. We assume that the diffusion coefficients are merely measurable
(possibly discontinuous) functions. In the case of Lipschitz continuous
coefficients we give the rate of $L^p$ approximation for every $p\geq1$. We
prove that if $D$ is a convex polyhedron then the rate is $O((\frac{\ln
n}n)^{1/2})$, and in the general case the rate is $O((\frac{\ln n}n)^{1/4})$.
