<p style='text-indent:20px;'>Bayesian inverse problems for inferring the unknown forcing and initial condition of Navier-Stokes equation play important roles in many practical areas. The computation cost of sampling the posterior probability measure can be exceedingly high. We develop the Finite Element Multilevel Markov Chain Monte Carlo (FE-MLMCMC) sampling method for approximating expectation with respect to the posterior probability measure of quantities of interest for a model problem of

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... vier-Stokes equation in the two dimensional periodic torus. We first consider the case where the forcing and the initial condition are bounded for all the realizations and depend linearly on a countable set of random variables which are uniformly distributed in a compact interval. We establish the essentially optimal convergence rate of the method and verify it numerically. The method follows from that developed in V. H. Hoang, Ch. Schwab and A. M. Stuart, Inverse problems, vol. 29, 2013 for inferring the coefficients of linear elliptic forward equations under the uniform prior probability measure. In the case of the Gaussian prior probability measure, numerical results, using the MLMCMC method developed for the Gaussian prior in V. H. Hoang, J. H. Quek and Ch. Schwab, Inverse problems, vol. 36, 2020, indicate the essentially optimal convergence rates. However, a rigorous theory for the MLMCMC sampling procedure is not available, due to the non-integrability with respect to the Gaussian prior of the theoretical finite element errors of the forward solvers that are available in the literature.</p>