A Finite Element Approach for the Dual Rudin-Osher-Fatemi Model and Its Nonoverlapping Domain Decomposition Methods

Chang-Ock Lee, Eun-Hee Park, Jongho Park

Abstract: We consider a finite element discretization for the dual Rudin--Osher--Fatemi model using a Raviart--Thomas basis for $H_0 ({div} ; \Omega)$. Since the proposed discretization has a splitting property for the energy functional, which is not satisfied for existing finite difference--based discretizations, it is more adequate for designing domain decomposition methods. In this paper, a primal domain decomposition method is proposed which resembles the classical Schur complement method for the second order elliptic problems, and it achieves $O(1/n^2)$ convergence. A primal-dual domain decomposition method based on the method of Lagrange multipliers on the subdomain interfaces is also considered. Local problems of the proposed primal-dual domain decomposition method can be solved at a linear convergence rate. Numerical results for the proposed methods are provided.