  Multilevel quadrature methods for parametric operator equations such as the
multilevel (quasi-) Monte Carlo method are closely related to the sparse tensor
product approximation between the spatial variable and the parameter. In this
article, we employ this fact and reverse the multilevel quadrature method via
the sparse grid construction by applying differences of quadrature rules to
finite element discretizations of increasing resolution. Besides being
algorithmically more efficient if the underlying quadrature rules are nested,
this way of performing the sparse tensor product approximation enables the easy
use of non-nested and even adaptively refined finite element meshes.
Especially, we present a rigorous error and regularity analysis of the fully
discrete solution, taking into account the effect of polygonal approximations
to a curved physical domain and the numerical approximation of the bilinear
form. Our results facilitate the construction of efficient multilevel
quadrature methods based on deterministic quadrature rules. Numerical results
in three spatial dimensions are provided to illustrate the approach.
