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.
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.
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Cahn-Hilliard
Cahn--Hilliard
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which has locally Lipschitz derivatives
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matrix-vector
matrix--vector
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weak
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matrix-vector formulation and it is utilised to prove
spatial discretisation. A new stability proof, based on this structure, combined with consistency bounds proves
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An
To demonstrate the advantages of the new stability analysis, an
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stability and
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general
generalised
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Cahn-Hilliard
Cahn--Hilliard-type