In order to solve the coupled system analytically or numerically, we need to give a description of our surface. There are several approaches we could take; parametric, graph, level set, phase field. We will consider the first two methods.
- Parametric approach - Mesh point will evolve with normal velocity leading to mesh degeneration. We aim to resolve this issue with the DeTurck trick.
- Graph approach - Although this assumption is restrictive in terms of modelling, it will enable us to prove local existence. The ideas and techniques will be a useful tool in understanding the general situation.
We parametrise the evolving closed curve using a stationary reference manifold,
A key problem with this approach is that our mesh points move with normal velocity resulting in mesh degeneration. As a result, this will produce large errors in our numerical schemes. To resolve this issue we will use the DeTurck trick, which is a reparametrisation that introduces tangential motion.Forced curve shortening flow with the DeTurck trick
Applying the DeTurck reparametrisation to curve shortening flow with forcing, we have
In local coordinates the equation becomes:
Discretising the spatial variables using linear finite elements we obtain the semi-discrete scheme
A suitable convergence result can be proved, however the constant in the error estimate will blow up as . The problem of convergence with remains open.