For the elliptic data assimilation problem, Gaussian priors defined directly on finite element spaces give discrete posterior means that contract to the ground truth at the optimal continuous-level rates when the mesh size and the number of samples are coupled as h ~ N^{-1/(2α+d)}.
Nickl , Bayesian non-linear statistical inverse problems , EMS press Berlin, 2023
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Posterior contraction rates of computational methods for Bayesian data assimilation
For the elliptic data assimilation problem, Gaussian priors defined directly on finite element spaces give discrete posterior means that contract to the ground truth at the optimal continuous-level rates when the mesh size and the number of samples are coupled as h ~ N^{-1/(2α+d)}.