The minimization of functionals which are formed by an L2-term and a Total Variation (TV) term play an important role in mathematical imaging with many applications in engineering, medicine and art. The TV term is well known to preserve sharp edges in images.

More precisely, we are interested in the minimization of a functional formed by a discrepancy term and a TV term. The first order derivative of the TV term involves a degenerate term which could happen in flat areas of an image. Many well known methods have been proposed to solve this problem.

In this thesis, we present a relaxed functional associated with the TV minimization problem. The relaxed functionals are well-posed and produce a sequence of solutions minimizing our original TV-functional. The relaxed functional results in an Iteratively Reweighted Least Squares method that approximates the TV minimization.

Considering the Euler-Lagrange equation, the minimizer of the relaxed functional is equivalent to the solution of a second order elliptic partial differential equation. We discretize this partial differential equation in the framework of Discontinuous Galerkin (DG) Finite Element Method (FEM) with linear functions on each element. Specifically, we consider the Symmetric Interior Penalty Galerkin method. The discretization leads to a system of linear equations.

The existence and uniqueness of the solution to the DG variational form of Discontinuous Galerkin and the discrete DG problem is studied, and a-priori error estimates are reported. The Discontinuous Galerkin Finite Element Method in combination with iteratively reweighted least squares method is implemented. Finally, numerical results are presented that demonstrate the accuracy of the numerical solution using the proposed methods.

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