  Let $V$ be a {\em periodic} potential on $\RR^3$ that is smooth everywhere
except at a discrete set $\maS$ of points, where it has singularities of the
form $Z/\rho^2$, with $\rho(x) = |x - p|$ for $x$ close to $p$ and $Z$ is
continuous, $Z(p) > -1/4$ for $p \in \maS$. We also assume that $\rho$ and $Z$
are smooth outside $\maS$ and $Z$ is smooth in polar coordinates around each
singular point. Let us denote by $\Lambda$ the periodicity lattice and set $\TT
:= \RR^3/ \Lambda$. In the first paper of this series \cite{HLNU1}, we obtained
regularity results in weighted Sobolev space for the eigenfunctions of the
Schr\"odinger-type operator $H = -\Delta + V$ acting on $L^2(\TT)$, as well as
for the induced $\vt k$--Hamiltonians $\Hk$ obtained by resticting the action
of $H$ to Bloch waves. In this paper we present two related applications: one
to the Finite Element approximation of the solution of $(L+\Hk) v = f$ and one
to the numerical approximation of the eigenvalues, $\lambda$, and
eigenfunctions, $u$, of $\Hk$. We give optimal, higher order convergence
results for approximation spaces defined piecewise polynomials. Our numerical
tests are in good agreement with the theoretical results.
