A revisit to DeTurck method on curve shortening flow with optimal error analysis

The curve shortening–DeTurck flow introduced by Elliott and Fritz (IMA J. Numer. Anal. 37(2): 543–603, 2017) is a reparameterization of the curve shortening flow via the DeTurck trick. For corresponding discrete schemes, although the $H^1$-error estimate has been established, the optimal $L^2$-error estimate remains open due to technical difficulties. In this paper, we prove optimal $L^2$-error estimates for linearized Euler and Crank–Nicolson discretizations combined with finite elements of degree $k\ge 1$ in space. Moreover, we provide an extrinsic approach to derive the curve shortening–DeTurck flow. Numerical experiments are presented to illustrate the mesh distribution properties and to verify the convergence rates in both space and time.