Energy stable symmetric interior penalty discontinuous Galerkin finite element for a growth Cahn-Hilliard equation

In this paper we devise and analyze a symmetric interior penalty (SIP) discontinuous Galerkin (DG) finite element (FE) method for a growth Cahn-Hilliard equation (GCH) with a polynomial potential and (general) nonlinear growth term, equipped with homogeneous Dirichlet boundary conditions. The original partial differential equation (PDE) system is rewritten by introducing an extra variable, and it is shown to be a gradient flow of an energy functional, that its solutions dissipate. The proposed first order convex splitting (CS) fully discrete scheme is shown to be energy stable with respect to a spatially discrete analogue of the continuous free energy of the system. At the fully discrete level the energy law is shown to hold unconditionally with respect to the temporal discretization parameter and conditionally with respect to the spatial discretization parameter.