Comparative computational analysis of the Cahn-Hilliard equation with emphasis on C1-continuous methods

The numerical treatment of the fourth-order Cahn-Hilliard equation is nonstandard. Using a Galerkin-method necessitates, for instance, piecewise smooth and globally C1-continuous basis functions or a mixed formulation. The latter is obtained introducing an auxiliary field which allows to rephrase the Cahn-Hilliard equation as a set of two coupled second-order equations. In view of this, the formulation in terms of the primal unknown appears to be a more intuitive and natural choice but requires a C1-continuous interpolation. Therefore, isogeometric analysis, using a spline basis, and natural element analysis are addressed in the present contribution. Mixed second-order finite element methods introducing the chemical potential or alternatively a nonlocal concentration as auxiliary field serve as references to which both higher-order methods are compared in terms of accuracy and efficiency.