A $C^0$ INTERIOR PENALTY DISCONTINUOUS GALERKIN METHOD FOR FOURTH ORDER TOTAL VARIATION FLOW
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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. We prove existence and uniqueness of a solution of the $C^0$ IPDG approximation by a nonlinear analogue of the Lax-Milgram Lemma. This requires to show that the nonlinear operator associated with the $C^0$ IPDG semilinear form is Lipschitz continuous and strongly monotone on bounded subsets of the underlying finite element space. 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.