A C0 interior penalty discontinuous Galerkin method for fourth‐order total variation flow. I: Derivation of the method and numerical results. Issue 4 (23rd February 2019)

• Record Type: Journal Article
• Authors: Bhandari, Chandi; Hoppe, Ronald H.W.; Kumar, Rahul
• Is Part Of: Numerical methods for partial differential equations. Volume 35:Issue 4(2019)
• Page Start: 1458
• Page End: 1476
• Publication Date: 2019-02-23
• Subjects: C0 interior penalty discontinuous Galerkin method -- fourth‐order total variation flow -- surface relaxation; Differential equations, Partial -- Numerical solutions -- Periodicals; 515.353
• DOI: 10.1002/num.22359
• Languages: English
• ISSNs: 0749-159X

Abstract

Abstract : 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.