  Optimal-order uniform-in-time $H^1$-norm error estimates are given for semi-
and full discretizations of mean curvature flow of surfaces in arbitrarily high
codimension.
  The proposed and studied numerical method is based on a parabolic system
coupling the surface flow to evolution equations for the mean curvature vector
and for the orthogonal projection onto the tangent space. The algorithm uses
evolving surface finite elements and linearly implicit backward difference
formulae.
  This numerical method admits a convergence analysis in the case of finite
elements of polynomial degree at least two and backward difference formulae of
orders two to five.
  Numerical experiments in codimension 2 illustrate and complement our
theoretical results.
