An evolving surface finite element method for the Cahn-Hilliard equation with a logarithmic potential

In this paper we study semi-discrete and fully discrete evolving surface finite element schemes for the Cahn-Hilliard equation with a logarithmic potential. Specifically we consider linear finite elements discretising space and backward Euler time discretisation. Our analysis relies on a specific geometric assumption on the evolution of the surface. Our main results are $L_{H^1}^2$ error bounds for both the semi-discrete and fully discrete schemes, and we provide some numerical results.