# Error estimates for general 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. 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 weak equation system is preserved by the matrix-vector formulation and it is utilised to prove optimal-order and uniform-in-time error estimates. An extension of the convergence results is given for general non-linear Cahn-Hilliard equations on evolving surfaces. The paper is concluded by a variety of numerical experiments.

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