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We are interested in the numerical reconstruction of a vector field withprescribed divergence and curl in a general domain of R 3 or R 2 , notnecessarily contractible. To this aim, we introduce some basic concepts offinite element exterieur calculus and rely heavily on recent results of P.Leopardi and A. Stern. The goal of the paper is to take advantage of the linksbetween usual vector calculus and exterior calculus and show the interest ofthe exterior calculus framework, without too much prior knowledge of thesubject. We start by describing the method used for contractible domains andits implementation using the FEniCS library (see fenicsproject.org). We thenaddress the problems encountered with non contractible domains and generalboundary conditions and explain how to adapt the method to handle these cases.Finally we give some numerical results obtained with this method, in dimension2 and 3.

Numerical solution of the div-curl problem by finite element exterior calculus