Variational convergence of discrete elasticae

IMA Journal of Numerical Analysis, vol. 42, pp. 300–332

Abstract

Abstract We discuss a discretization of the Euler–Bernoulli bending energy and of Euler elasticae under clamped boundary conditions by polygonal lines. We show Hausdorff convergence of the set of almost minimizers of the discrete bending energy to the set of smooth Euler elasticae under mesh refinement in (i) the $W^{1,\infty }$-topology for piecewise-linear interpolation; and in (ii) the $W^{2,p}$-topology, $p \in [2,\infty [$, using a suitable smoothing operator to create $W^{2,p}$-curves from polygons.

Authors 3

1. Sebastian Scholtes Aachen Institute for Mathematics

   RWTH Aachen University

   Affiliation as printed

   Institute for Mathematics, RWTH Aachen University, Templergraben 55, D-52062 Aachen, Germany

   RWTH Aachen University

2. H. Ralph Schumacher corresponding Aachen Institute for Mathematics

   RWTH Aachen University

   Affiliation as printed

   Institute for Mathematics, RWTH Aachen University, Templergraben 55, D-52062 Aachen, Germany

   RWTH Aachen University

3. Max Wardetzky

   University of Göttingen

   Affiliation as printed

   Institute for Numerical and Applied Mathematics, University Göttingen, Lotzestraße 16–18, D-37083 Göttingen, Germany

   University of Gottingen;