We consider the setting where there is an obstacle lying above the graph in the coupled system (coupled system) . To do this we reformulate our original motion by mean curvature problem into a variational inequality
which can also be considered as the minimisation of
where K is a convex set constraining the graph to lie below the obstacle.To solve this we interpolate the initial curve with cubic splines and then use a relaxtion algorithm to minimise at each time step.
Considering only a constant forcing term in the upward normal direction, that is , Running this with 400 time steps and nodal width , . We see that the curve reaches an equilibrium where the downward forcing of the obstance and mean curvature is enough to overcome to upward forcing on the curve.
We can also extend this from constant forcing to include a diffsive scalar field on the surface (link to coupled system equations) this is discritized using linear finite elements.
We can compare this to a pure finite element scheme for the coupled system with no obstacles