This work studies the field prediction and smoothing problems, where the spatio-temporal field in 2-D is described by a stochastic dynamical system and observed by a number of spatially deployed sensors. We adopt a finite-element technique to approximate the field dynamics with piece-wise Gaussian functions, leading to a high-dimensional linear stochastic system. By exploiting its sparsity, a local covariance intersection-based filter and smoother are developed in each sensor only for a moderate number of state variables via communications with nearby sensors. Such a cooperative scheme is both communication and computation efficient. We prove the uniform stability of the local filter and smoother under mild conditions, and validate their effectiveness on two application examples: the temperature prediction of a metal rod and the source localization of a PM<inline-formula><tex-math notation="LaTeX">_{2.5}</tex-math></inline-formula> field with a real dataset in a city of China.