Finite element approximation of data-driven problems in conductivity

This paper is concerned with the finite element discretization of the data driven approach according to Kirchdoerfer and Ortiz (Comput Methods Appl Mech Eng 304:81-101, 2016) 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 Conti et al. (Arch Ration Mech Anal 229(1):79-123, 2018) carries over to the finite element discretization as long as H(div)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H({\text {div}})$$\end{document}-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 Conti et al., 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 three examples and it turns out that the "classical" alternating projection method according to Kirchdoerfer and Ortiz is superior w.r.t. the ratio of accuracy and computational time.