Quasi-Monte Carlo and discontinuous Galerkin

TL;DR

This paper develops tailored quasi-Monte Carlo methods for non-conforming discontinuous Galerkin approximations of elliptic PDEs with random coefficients, demonstrating comparable convergence rates to continuous finite element methods.

Contribution

It introduces new parametric regularity bounds for DG methods and proves QMC convergence rates match those of continuous finite elements for PDEs with random inputs.

Findings

QMC cubatures achieve optimal convergence rates for DG approximations.

Parametric regularity bounds for DG are established and applicable to other methods.

Numerical results confirm the theoretical convergence rates and effectiveness.

Abstract

In this study, we consider the development of tailored quasi-Monte Carlo (QMC) cubatures for non-conforming discontinuous Galerkin (DG) approximations of elliptic partial differential equations (PDEs) with random coefficients. We consider both the affine and uniform and the lognormal models for the input random field, and investigate the use of QMC cubatures to approximate the expected value of the PDE response subject to input uncertainty. In particular, we prove that the resulting QMC convergence rate for DG approximations behaves in the same way as if continuous finite elements were chosen. Notably, the parametric regularity bounds for DG, which are developed in this work, are also useful for other methods such as sparse grids. Numerical results underline our analytical findings.