Quasi-optimal error estimate for the approximation of the elastic flow of inextensible curves

Abstract In previous research numerical schemes for the approximation of inextensible elastic curves have been proposed, which approximate the curve with piecewise cubic functions and impose the inextensibility constraint $\vert z_{x} \vert ^{2} = 1$ only at the nodes of the underlying interval dissection. Numerical simulations for these schemes show linear convergence in $H^{2}$, though quadratic convergence is the optimal expected rate for cubic splines. In this paper a new discretization of the elastic flow of inextensible curves, which imposes the constraint, not just at the endpoints, but also at the midpoints of each subinterval, is devised and quasi-optimal quadratic convergence of the corresponding semidiscrete problem is proved. Further, a fully discrete time-stepping scheme that incorporates this constraint is proposed and unconditional stability and convergence of the discrete scheme are proved. Finally, some numerical simulations are used to verify the obtained results experimentally.