Multilevel quadrature for elliptic parametric partial differential equations on non-nested meshes

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 stochastic variable. 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 different resolution. Besides being more efficient if the underlying quadrature rules are nested, this way of performing the sparse tensor product approximation enables the use of non-nested and even adaptively refined finite element meshes. Especially, the multilevel quadrature is non-intrusive and allows the use of standard finite element solvers. Numerical results are provided to illustrate the approach.