Sufficient conditions for QMC analysis of finite elements for parametric differential equations

TL;DR

This paper establishes conditions under which Quasi-Monte Carlo methods achieve fast, dimension-independent convergence rates for finite element discretizations of parametric diffusion equations with Gevrey-regular random parameters.

Contribution

It provides a set of verifiable assumptions ensuring QMC error bounds for various finite element methods applied to parametric PDEs with Gevrey-regular randomness.

Findings

QMC achieves faster-than-Monte Carlo convergence under certain regularity conditions.

Assumptions verified for conforming finite elements, mixed methods, and hybridizable discontinuous Galerkin schemes.

Numerical experiments confirm the theoretical convergence rates and importance of flux approximation.

Abstract

Parametric regularity of discretizations of flux vector fields satisfying a balance law is studied under some assumptions on a random parameter that links the flux with an unknown primal variable (often through a constitutive law). In the primary example of the stationary diffusion equation, the parameter corresponds to the inverse of the diffusivity. The random parameter is modeled here as a Gevrey-regular random field. Specific focus is on random fields expressible as functions of countably infinite sequences of independent random variables, which may be uniformly or normally distributed. Quasi-Monte Carlo (QMC) error bounds for some quantity of interest that depends on the flux are then derived using the parametric regularity. It is shown that the QMC method achieves a dimension-independent, faster-than-Monte Carlo convergence rate if the quantity of interest depends continuously on the primal variable, its flux, or its gradient. A series of assumptions are introduced with the goal of encompassing a broad class of discretizations by various finite element methods. The assumptions are verified for the diffusion equation discretized using conforming finite elements, mixed methods, and hybridizable discontinuous Galerkin schemes. Numerical experiments confirm the analytical findings, highlighting the role of accurate flux approximation in QMC methods.