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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

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