This paper introduces Firedrake, a new automatic system to generate code and solve partial differential equations using finite element methods. This capability is a core need of many models, and consequently a source of significant redundant software development effort. Because it does not prescribe a particular set of equations, the Firedrake software is applicable to a wide range of geoscientific models. Firedrake demonstrates remarkable computational efficiency, scaling beyond 12,000 computing cores. It is also free-libre open source software, contributing to improvements in scientific computational replicability and reproducibility.