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We devised a fully discrete scheme for finding approximate solutions

 x : [0,T] \times S^1 \to \mathbb{R}^2, \qquad c : [0,T] \times S^1 \to \mathbb{R}, \qquad \lambda : [0,T] \to \mathbb{R}

to the system of equations

 \nu \cdot \dot{x} = - k_s H + k_b \, (\Delta_\Gamma H + \tfrac{1}{2} H^3) + \lambda + f({c}),
 \tau \cdot \dot{x} = \frac{1}{\alpha} \, \frac{x_p \cdot x_{pp}}{|x_p|^3},
 \dot{c} = k_c \, \Delta_\Gamma {c} - {c} \, \nabla_\Gamma \cdot \dot{x} + \nabla_\Gamma \cdot (c \, \tau \, (\tau \cdot \dot{x})),
 0 = \int_{S^1} \nu \cdot \dot{x} \, dp.

Prior work. Several schemes for equations of this kind have been proposed in the literature, all of which introduce an auxiliary variable for the mean curvature H determined using H \nu = - \Delta_\Gamma x. This makes it difficult to impose the DeTurck tangential velocity and increases the computational effort.

Our approach. We instead propose to use the identity H = - \frac{x_p^\perp \cdot x_{pp}}{|x_p|^3} and use H^2-conforming finite elements to discretise the resulting fourth-order equation. Apart from avoiding auxiliary variables, this allows for fourth-order convergence in terms of the number of spatial degrees of freedom.
We use a semi-implicit time-stepping scheme which only requires the solution of linear systems. The downside of this approach is that it only yields first-order convergence.

Assessing the DeTurck reparametrisation.

The following clips show the mesh evolution for two different values of $\alpha$. The merits of the DeTurck reparametrisation (left series) compared to a purely normal flow (right series) are obvious.

\alpha = 10^{-4}
(strong tangential redistribution)

\alpha = 10^{4}
(virtually no tangential redistribution)