We study the gradient flow for the total variation
functional, which arises in image processing and geometric applications. We propose a variational inequality weak formulation for the gradient flow,
and establish well-posedness of the problem by the energy method.
The main idea of our approach is to exploit the relationship between
the regularized gradient flow (characterized by a small positive parameter
ε, see (1.7)) and the minimal surface flow [21]
and the prescribed mean curvature flow [16].
Since our approach is constructive
and variational, finite element methods can be naturally
applied to approximate weak solutions of the limiting gradient
flow problem. We propose a fully discrete finite element method
and establish convergence to
the regularized gradient flow problem as h,k → 0, and to the
total variation gradient flow problem as h,k,ε → 0
in general cases.
Provided that the regularized gradient flow problem possesses
strong solutions, which is proved possible if the datum functions
are regular enough, we establish practical a priori error estimates
for the fully discrete finite element solution, in particular, by focusing
on the dependence of the error bounds on the
regularization parameter ε. Optimal order error bounds are
derived for the numerical solution under the mesh
relation k = O(h2). In particular, it is shown that
all error bounds depend on $\frac{1}{\varepsilon}$ only
in some lower polynomial order for small ε.