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A C 0 interior penalty discontinuous Galerkin method for fourth‐order total variation flow. I: Derivation of the method and numerical results
Author(s) -
Bhandari Chandi,
Hoppe Ronald H.W.,
Kumar Rahul
Publication year - 2019
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22359
Subject(s) - mathematics , discretization , discontinuous galerkin method , uniqueness , nonlinear system , galerkin method , penalty method , mathematical analysis , regularization (linguistics) , continuation , superconvergence , finite element method , mathematical optimization , physics , quantum mechanics , artificial intelligence , computer science , thermodynamics , programming language
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.