Finite element approximation of an H1 gradient flow of a double-well potential with bending energy

In this paper we consider the discretization error in space and time of an H¹ gradient flow for an energy integral where the energy density is given by the sum of a double-well potential term and a bending energy term. We show that the problem is equivalent to a nonlinear heat equation with nonlocal nonlinearity and adapt the standard error analysis theory developed for the nonlinear heat equation to our case. In doing so we bound the discretization error in terms of the mesh size and time step as well as energy parameters. In particular, we carefully track how the size of the bending energy affects the error bounds.