Numerical Simulation of 4th Order Total Variation Flow Problem by using C^0 IPDG Method

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2018-05

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Abstract

This dissertation is devoted to the the numerical solution of the regularized fourth order total variation flow problem in material science representing surface relaxation below the roughening temperature. Based on regularization and a scaling in time and space, the problem is discretized implicitly in time by the backward Euler scheme and discretized in space by C0 Interior Penalty Discontinuous Galerkin (C0IPDG) method. In particular, it is showed that at each time instant the C0IPDG approximation represents the necessary and sufficient optimality condition for the minimization of an associated proper convex, coercive, and lower semi-continuous objective functional. The main results are a priori error estimates of the global discretization error in a mesh dependent C0IPDG-norm and the L2-norm. A documentation of numerical results is provided illustrating the performance of the C0IPDG method and predictor corrector continuation strategies.

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Keywords

Surface relaxation, Galerkin approximation, C 0 Interior Penalty, Discontinuous Galerkin Approximation, Mixed finite element methods

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