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Charles M. Elliott, Sheila SmithemanNumerical Analysis of the TV Regularisations and H-1 Fidelity Model for Decomposing an Image in to Cartoon plus TextureThe Osher-Sole-Vese (OSV) Model [10], which is the gradient flow of an energy consitsting of the total variation functional plus an H-1 fidelity term, is studied. In this paper, we build on the analysis of the OSV model which we gave in [6]. We introduce backward Euler finite element approximation to a regularised version of the OSV initial boundary value problem (IBVP) and to a weak formulation of the original problem. Well-posedness and unconditional Lyapunov stability of these fully discrete schemes is proved. Convergence results as the spatial mesh parameter, time step size and the regularisation parameter tend to 0 are proved. Rates of convergence as the time step size and the regularisation parameter tend to 0 are found. The existence, uniqueness and Lyapunov stability of a solution to a linearly implicit finite element approximation to the regularised version of the OSV IBVP is also proved.Bibliographical note: Submitted to Communications on Pure and Applied Analysis |