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

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. KEYWORDS C 0 interior penalty discontinuous Galerkin method, fourth-order total variation flow, surface relaxation

A C0 interior penalty discontinuous Galerkin Method for fourth‐order total variation flow. II: Existence and uniqueness

“…As has been shown in Bhandari and coworkers , we have where D 2 u m is the 2 × 2 matrix of second partial derivatives of u m and the matrix is given by …”

“…Following the general approach for DG approximations of second‐order elliptic boundary value problems, in Bhandari and coworkers we have derived the following C 0 IPDG approximation of Equation (3.7): Find such that for all v h ∈ V h it holds …”

“…We note that similar fourth‐order TVF problems occur in image recovery. For more details we refer to Bhandari and coworkers and the references therein.…”

“…At each time‐step, this amounts to the solution of 2 second‐order elliptic PDEs by standard Lagrangian finite elements with respect to a triangulation of the computational domain Ω . On the other hand, a C 0 Interior Penalty Discontinuous Galerkin (C 0 IPDG) method has been developed and implemented in Bhandari and coworkers . The advantage of the C 0 IPDG approach is that it directly applies to the fourth‐order problem and thus only requires the numerical solution of one equation by using the same Lagrangian finite elements as in the mixed method.…”

“…The paper is organized as follows: After some basic notations from matrix analysis and Lebesgue and Sobolev spaces presented in Section 2, in Section 3 we recall the C 0 IPDG approximation of the implicity in time discretized, regularized, and scaled fourth‐order TVF problem from Bhandari and coworkers . Section 4 is devoted to a proof of the existence and uniqueness of a solution of the C 0 IPDG approximation by an application of the nonlinear version of the Lax‐Milgram Lemma.…”