Error estimates for generalised non-linear Cahn-Hilliard equations on evolving surfaces

In this paper, we consider the Cahn--Hilliard equation on evolving surfaces with prescribed velocity and a general non-linear potential which has locally Lipschitz derivatives. High-order evolving surface finite elements are used to discretise the weak equation system in space, and a modified matrix--vector formulation for the semi-discrete problem is derived. The anti-symmetric structure of the equation system is preserved by the spatial discretisation. A new stability proof, based on this structure, combined with consistency bounds proves optimal-order and uniform-in-time error estimates. To demonstrate the advantages of the new stability analysis, an extension of the stability and convergence results is given for generalised non-linear Cahn--Hilliard-type equations on evolving surfaces. The paper is concluded by a variety of numerical experiments.