Variational convergence of discrete elasticae

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.