Total Variation Minimization with Finite Elements: Convergence and Iterative Solution

Abstract

The numerical solution of a convex minimization problem involving the nonsmooth total variation norm is analyzed. Consistent finite element discretizations that avoid regularizations lead to simple convergence proofs in the case of piecewise affine, globally continuous finite elements. For the approximation with piecewise constant finite elements it is proved that convergence to the exact solution cannot be expected in general. The iterative solution is based on a regularized $L^2$ flow of the energy functional, and convergence of the iteration to a stationary point is proved under a moderate constraint on the time-step size. The extension of the techniques to an energy functional that involves a negative order term is discussed. Numerical experiments that illustrate the theoretical results are presented.