Multilevel Quadrature for Elliptic Parametric Partial Differential Equations in Case of Polygonal Approximations of Curved Domains

Multilevel quadrature methods for parametric operator equations such as the multilevel (quasi-) Monte Carlo method resemble a sparse tensor product approximation between the spatial variable and the parameter. We employ this fact to reverse the multilevel quadrature method 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 nonnested and even adaptively refined finite element meshes. We moreover provide a rigorous error and regularity analysis addressing the variational crimes of using polygonal approximations of curved domains and numerical quadrature of the bilinear form. Our results facilitate the construction of efficient multilevel quadrature methods based on deterministic high order quadrature rules for the stochastic parameter. Numerical results in three spatial dimensions are provided to illustrate the approach.