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In this paper, we propose a novel framework for field estimation in a wireless sensor network (WSN). The fundamental problem of estimating field values at locations where no WSN measurements are available is tackled by including a physical field model in the form of a partial differential equation (PDE). If the PDE is discretized in the spatiotemporal domain by use of the finite element method (FEM), then the physical field model reduces to a set of linear equations that can be elegantly combined with the WSN field measurements in a constrained optimization problem. In contrast to existing approaches, we do not require the driving source function or the locations of point sources to be known. Instead, we assume limited prior knowledge on the nature of the field and/or source functions, such as a sparsity or nonnegativity prior, for obtaining a unique solution of the otherwise underdetermined problem of joint field and source estimation. Within the proposed framework, we derive a cooperative estimation algorithm for static 2-D fields governed by a Poisson PDE. Simulation results illustrate that a significant improvement in field estimation accuracy can be obtained, compared to the cases when only WSN measurements (without a physical model) or only the FEM (without WSN measurements) are used.