Abstract: We study the consistency of the estimator in a spatial regression with partial differential equation (PDE) regularization. This new smoothing technique allows us to accurately estimate spatial fields over complex two-dimensional domains, starting from noisy observations. The regularizing term involves a PDE that formalizes problem-specific information about the phenomenon at hand. In contrast to classical smoothing methods, the solution to the infinite-dimensional estimation problem cannot be computed analytically. An approximation is obtained using the finite-element method, considering a suitable triangulation of the spatial domain. We first consider the consistency of the estimator in the infinite-dimensional setting. We then study the consistency of the finite-element estimator resulting from the approximated PDE. We study the bias and variance of the estimators with respect to the sample size and the value of the smoothing parameter. Lastly, simulation studies provide numerical evidence of the rates derived for the bias, variance, and mean square error.

Key words and phrases: Functional data analysis, smoothing, spatial statistics.