  A proof of convergence is given for semi- and full discretizations of mean
curvature flow of closed two-dimensional surfaces. The numerical method
proposed and studied here combines evolving finite elements, whose nodes
determine the discrete surface like in Dziuk's method, and linearly implicit
backward difference formulae for time integration. The proposed method differs
from Dziuk's approach in that it discretizes Huisken's evolution equations for
the normal vector and mean curvature and uses these evolving geometric
quantities in the velocity law projected to the finite element space. 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. The error analysis combines stability estimates and consistency
estimates to yield optimal-order $H^1$-norm error bounds for the computed
surface position, velocity, normal vector and mean curvature. The stability
analysis is based on the matrix--vector formulation of the finite element
method and does not use geometric arguments. The geometry enters only into the
consistency estimates. Numerical experiments illustrate and complement the
theoretical results.
