Quasi-Monte Carlo finite element approximation of the Navier-Stokes equations with initial data modeled by log-normal random fields

TL;DR

This paper develops a rigorous numerical method combining finite elements, QMC, and Karhunen-Loève expansion to efficiently approximate the expected behavior of Navier-Stokes solutions with uncertain initial data modeled by log-normal random fields.

Contribution

It introduces a novel combination of finite element discretization, lattice-based QMC, and stochastic expansion for analyzing Navier-Stokes equations with uncertain initial conditions.

Findings

Error decays as O(N^{-1+δ}) with the number of sampling points N.

Method is independent of the high-dimensionality of the parameter space.

Provides rigorous error analysis for the approximation of expected functionals.

Abstract

In this paper, we analyze the numerical approximation of the Navier-Stokes problem over a bounded polygonal domain in $\mathbb{R}^2$, where the initial condition is modeled by a log-normal random field. This problem usually arises in the area of uncertainty quantification. We aim to compute the expectation value of linear functionals of the solution to the Navier-Stokes equations and perform a rigorous error analysis for the problem. In particular, our method includes the finite element, fully-discrete discretizations, truncated Karhunen-Lo\'eve expansion for the realizations of the initial condition, and lattice-based quasi-Monte Carlo (QMC) method to estimate the expected values over the parameter space. Our QMC analysis is based on randomly-shifted lattice rules for the integration over the domain in high-dimensional space, which guarantees the error decays with $\mathcal{O}(N^{-1+\delta})$, where $N$ is the number of sampling points, $\delta>0$ is an arbitrary small number, and the constant in the decay estimate is independent of the dimension of integration.