# Variational Convergence of Discrete Elasticae

## TL;DR

This paper proves that polygonal approximations of Euler elasticae converge to the smooth curves as the mesh is refined, in specific topologies, using discretization and smoothing techniques.

## Contribution

It establishes Hausdorff convergence of discrete elasticae to smooth elasticae under mesh refinement in multiple function space topologies.

## Key findings

- Hausdorff convergence in $W^{1, ext{infinity}}$-topology
- Hausdorff convergence in $W^{2,p}$-topology for $p o ext{infinity}$
- Use of smoothing operators to connect polygons to $W^{2,p}$-curves

## Abstract

We discuss a discretization by polygonal lines of the Euler-Bernoulli bending energy and of Euler elasticae under clamped boundary conditions. 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.