  Let $\Omega \subset \RR^d$, $d \geqslant 1$, be a bounded domain with
piecewise smooth boundary $\partial \Omega $ and let $U$ be an open subset of a
Banach space $Y$. Motivated by questions in "Uncertainty Quantification," we
consider a parametric family $P = (P_y)_{y \in U}$ of uniformly strongly
elliptic, second order partial differential operators $P_y$ on $\Omega$. We
allow jump discontinuities in the coefficients. We establish a regularity
result for the solution $u: \Omega \times U \to \RR$ of the parametric,
elliptic boundary value/transmission problem $P_y u_y = f_y$, $y \in U$, with
mixed Dirichlet-Neumann boundary conditions in the case when the boundary and
the interface are smooth and in the general case for $d=2$. Our regularity and
well-posedness results are formulated in a scale of broken weighted Sobolev
spaces $\hat\maK^{m+1}_{a+1}(\Omega)$ of Babu\v{s}ka-Kondrat'ev type in
$\Omega$, possibly augmented by some locally constant functions. This implies
that the parametric, elliptic PDEs $(P_y)_{y \in U}$ admit a shift theorem that
is uniform in the parameter $y\in U$. In turn, this then leads to
$h^m$-quasi-optimal rates of convergence (i.e. algebraic orders of convergence)
for the Galerkin approximations of the solution $u$, where the approximation
spaces are defined using the "polynomial chaos expansion" of $u$ with respect
to a suitable family of tensorized Lagrange polynomials, following the method
developed by Cohen, Devore, and Schwab (2010).
