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Variational Convergence of Discrete Elasticae

Analysis, Dynamics, and Applications Seminar

Variational Convergence of Discrete Elasticae
Series: Analysis, Dynamics, and Applications Seminar
Location: MATH 402
Presenter: Henrik Schumacher, , Department of Mathematics, RWTH Aachen University, Germany

In this talk, I am going to give a brief exposition of my recent work on discrete Euler elastica (joint work with Sebastian Scholtes and Max Wardetzky).  Discrete Euler elastica are critical points of the Euler-Bernoulli bending energy subject to inextensibility constraints and clamped boundary conditions.  A popular discrete bending energy for polygonal lines consists of a weighted sum of squared turning angles. We show that, under mesh refinement, a certain subset of almost minimizers of this energy (subject to inextensibility constraints and clamped boundary conditions) converges to the set of smooth minimizers:  When using piecewise-linear interpolation as reconstruction operator, we obtain Hausdorff-convergence in the space of Lipschitz curves. However, the proof hinges on a reconstruction operator that embeds discrete curves into $\operatorname{BV}^3$ (the space of curves whose third derivative is a measure of bounded variation). Modulo this operator, we even obtain Hausdorff-convergence in the Sobolev space $W^{2,p}$, $p <\infty$. This means that curvature information derived from discrete minimizers can indeed be trusted; since the discretization is not conforming, this is quite remarkable.