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

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