GMD Geoscientific Model Development GMD Geosci. Model Dev. 1991-9603 Copernicus Publications Göttingen, Germany 10.5194/gmd-6-2099-2013 Automating the solution of PDEs on the sphere and other manifolds in FEniCS 1.2 Rognes M. E. 1 Ham D. A. https://orcid.org/0000-0001-9545-9110 2 3 Cotter C. J. 2 McRae A. T. T. 2 4 Center for Biomedical Computing, Simula Research Laboratory, P.O. Box 134, 1325 Lysaker, Norway Department of Mathematics, Imperial College London, London SW7 2AZ, UK Department of Computing, Imperial College London, London SW7 2AZ, UK The Grantham Institute for Climate Change, Imperial College London, London SW7 2AZ, UK 17 12 2013 6 6 2099 2119 Copyright: © 2013 M. E. Rognes et al. 2013 This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/ This article is available from https://gmd.copernicus.org/articles/6/2099/2013/gmd-6-2099-2013.html The full text article is available as a PDF file from https://gmd.copernicus.org/articles/6/2099/2013/gmd-6-2099-2013.pdf

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.

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