This paper deals with the distributed implementation of a recently proposed algorithm for the estimation of static fields. The algorithm combines wireless sensor network (WSN) field measurements with a physical field model in the form of a partial differential equation (PDE), such that the field can be estimated at locations different from the WSN sensor node locations. By discretizing the PDE using the finite element method (FEM), the physical field model reduces to a highly sparse linear system of equations. It is shown how this FEM-induced sparsity pattern can be exploited in the design of a distributed implementation such as to minimize the communication effort and data storage required in the WSN. Simulation results illustrate that a significant improvement in field estimation accuracy can be obtained, compared to the case when only WSN measurements (without a physical model) are used.