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Elastic flow for closed curves can involve significant deformations.Mesh-based approximation schemes require tangentially redistributing verticesfor long-time computations. We present and analyze a method that uses theDirichlet energy for this purpose. The approach effectively also penalizes thelength of the curve, and equilibrium shapes are equivalent to stationary pointsof the elastic energy augmented with the length functional. Our numericalmethod is based on linear parametric finite elements. Following the lines of KDeckelnick, and G Dziuk (Math Comp 78, 266 (2009), 645-671) we proveconvergence and establish error estimates, noting that the addition of theDirichlet energy simplifies the analysis in comparison with the lengthfunctional. We also present a simple semi-implicit time discretization anddiscuss some numerical result that support the theory.

Convergence of a scheme for elastic flow with tangential mesh movement