  This paper is concerned with the finite element discretization of the data
driven approach according to arXiv:1510.04232 for the solution of PDEs with a
material law arising from measurement data. To simplify the setting, we focus
on a scalar diffusion problem instead of a problem in elasticity. It is proven
that the data convergence analysis from arXiv:1708.02880 carries over to the
finite element discretization as long as $H(\mathrm{div})$-conforming finite
elements such as the Raviart-Thomas element are used. As a corollary,
minimizers of the discretized problems converge in data in the sense of
arXiv:1708.02880, as the mesh size tends to zero and the approximation of the
local material data set gets more and more accurate. We moreover present
several heuristics for the solution of the discretized data driven problems,
which is equivalent to a quadratic semi-assignment problem and therefore
NP-hard. We test these heuristics by means of two examples and it turns out
that the "classical" alternating projection method according to
arXiv:1510.04232 is superior w.r.t. the ratio of accuracy and computational
time.
