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Multilevel quadrature methods for parametric operator equations such as themultilevel (quasi-) Monte Carlo method are closely related to the sparse tensorproduct approximation between the spatial variable and the parameter. In thisarticle, we employ this fact and reverse the multilevel quadrature method viathe sparse grid construction by applying differences of quadrature rules tofinite element discretizations of increasing resolution. Besides beingalgorithmically more efficient if the underlying quadrature rules are nested,this way of performing the sparse tensor product approximation enables the easyuse of non-nested and even adaptively refined finite element meshes.Especially, we present a rigorous error and regularity analysis of the fullydiscrete solution, taking into account the effect of polygonal approximationsto a curved physical domain and the numerical approximation of the bilinearform. Our results facilitate the construction of efficient multilevelquadrature methods based on deterministic quadrature rules. Numerical resultsin three spatial dimensions are provided to illustrate the approach.

Multilevel quadrature for elliptic parametric partial differential equations in case of polygonal approximations of curved domains