Quasi-optimal error estimates for the approximation of stable stationary states of the elastic energy of inextensible curves

We establish local existence and a quasi-optimal error estimate for piecewise cubic minimizers to the bending energy under a discretized inextensibility constraint. In previous research a discretization is used where the inextensibility constraint is only enforced at the nodes of the discretization. We show why this discretization leads to suboptimal convergence rates and we improve on it by also enforcing the constraint in the midpoints of each subinterval. We then use the inverse function theorem to prove existence and an error estimate for stationary states of the bending energy that yields quasi-optimal convergence. We use numerical simulations to verify the theoretical results experimentally.