Automating the solution of PDEs on the sphere and other manifolds in FEniCS 1.2

Differential equations posed over immersed manifolds are of particular importance in studying geophysical flows; for instance, ocean and atmosphere simulations crucially rely on the capability to solve equations over the sphere. This paper presents the extension of the FEniCS software components to the automated solution of finite element formulations of differential equations defined over general, immersed manifolds. We describe the implementation and, in particular detail, how the required extensions essentially reduce to the extension of the FEniCS form compiler to cover this case. The resulting implementation has all the properties of the FEniCS pipeline and we demonstrate its flexibility by an extensive range of numerical examples covering a number of geophysical benchmark examples and test cases. The results are all in agreement with the expected values. The description here relates to DOLFIN/FEniCS 1.2.