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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 highcodimension.  The proposed and studied numerical method is based on a parabolic systemcoupling the surface flow to evolution equations for the mean curvature vectorand for the orthogonal projection onto the tangent space. The algorithm usesevolving surface finite elements and linearly implicit backward differenceformulae.  This numerical method admits a convergence analysis in the case of finiteelements of polynomial degree at least two and backward difference formulae oforders two to five.  Numerical experiments in codimension 2 illustrate and complement ourtheoretical results.

A convergent finite element algorithm for mean curvature flow in arbitrary codimension