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.

翻译：本文关注基于arXiv:1510.04232数据驱动方法对含有测量数据材料定律的偏微分方程进行有限元离散求解。为简化研究背景，我们聚焦于标量扩散问题而非弹性问题。研究证明，只要使用$H(\mathrm{div})$协调的有限元（如Raviart-Thomas元），arXiv:1708.02880中的数据收敛分析即可推广至有限元离散情形。作为推论，当网格尺寸趋近于零且局部材料数据集逼近精度逐步提高时，离散问题极小值的解在arXiv:1708.02880定义的数据意义上收敛。此外，我们提出若干求解离散化数据驱动问题的启发式算法——该问题等价于二次半分配问题，属于NP难问题。通过两个数值算例检验这些启发式方法，结果表明基于arXiv:1510.04232的"经典"交替投影法在精度与计算时间比值方面表现更优。