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

TL;DR

This paper develops a multilevel quadrature method for elliptic parametric PDEs on polygonal approximations of curved domains, enabling efficient computation with adaptive meshes and providing rigorous error analysis.

Contribution

It introduces a reversed multilevel quadrature approach using sparse grid construction that handles non-nested and adaptive finite element meshes effectively.

Findings

Algorithmic efficiency with nested quadrature rules

Rigorous error analysis including polygonal domain approximation effects

Numerical validation in three spatial dimensions

Abstract

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.