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We present a two-dimensional conforming virtual element method for thefourth-order phase-field equation. Our proposed numerical approach to thesolution of this high-order phase-field (HOPF) equation relies on the design ofan arbitrary-order accurate, virtual element space with $C^1$ globalregularity. Such regularity is guaranteed by taking the values of the virtualelement functions and their full gradient at the mesh vertices as degrees offreedom. Attaining high-order accuracy requires also edge polynomial moments ofthe trace of the virtual element functions and their normal derivatives. Inthis work, we detail the scheme construction, and prove its convergence byderiving error estimates in different norms. A set of representative test casesallows us to assess the behavior of the method.

A $C^1$-conforming arbitrary-order two-dimensional virtual element method for the fourth-order phase-field equation