Convergence of isoparametric BGN methods for mean curvature flow of closed surfaces with high-order finite elements

We prove the convergence of the isoparametric Barrett–Garcke–Nürnberg (BGN) method for mean curvature flow of closed surfaces. For finite elements of degree $k\ge 6$ and time steps $\tau =O(h^2)$, the numerical surfaces converge to the exact surfaces in the manifold distance with order $h^2$. The proof relies on two ingredients. First, we construct approximation flows by an expansion in time, whose interpolations satisfy the BGN method with an improved defect. Second, we derive a refined estimate for the tangential error from an algebraic identity for tangential test functions.