Super-approximation and optimal $L^2$ convergence of the stabilized BGN method with piecewise linear parametric finite elements for curve-shortening flow

We develop a bihomotopy argument to estimate the discrete material time derivatives of consistency functionals. Combined with summation-by-parts, this argument yields a directional super-convergence estimate for the consistency contribution when tested with the normal error velocity. Together with the super-approximation of the reversely weighted discrete normal, these techniques establish $L_t^{\mathrm{\infty }}{H}_x^1$ super-convergence of the projection error and optimal $L_t^{\mathrm{\infty }}{L}_x^2$ convergence of the trajectory error of a stabilized BGN method for curve-shortening flow with piecewise linear parametric finite elements.