A C0 interior penalty discontinuous Galerkin method for fourth-order total variation flow. I: Derivation of the method and numerical results

We consider the numerical solution of a fourth-order total variation flow problem representing surface relaxation below the roughening temperature. Based on a regularization and scaling of the nonlinear fourth-order parabolic equation, we perform an implicit discretization in time and a C-0 Interior Penalty Discontinuous Galerkin (C(0)IPDG) discretization in space. The C(0)IPDG approximation can be derived from a mixed formulation involving numerical flux functions where an appropriate choice of the flux functions allows to eliminate the discrete dual variable. The fully discrete problem can be interpreted as a parameter dependent nonlinear system with the discrete time as a parameter. It is solved by a predictor corrector continuation strategy featuring an adaptive choice of the time step sizes. A documentation of numerical results is provided illustrating the performance of the C(0)IPDG method and the predictor corrector continuation strategy. The existence and uniqueness of a solution of the C(0)IPDG method will be shown in the second part of this paper.