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

翻译：我们提出了一种用于四阶相场方程的二维协调虚拟元方法。针对该高阶相场（HOPF）方程，我们提出的数值解法基于一种具有$C^1$全局正则性的任意阶精确虚拟元空间的设计。这种正则性通过将虚拟元函数及其完整梯度在网格顶点处的取值作为自由度来保证。实现高阶精度还需利用虚拟元函数迹及其法向导数的边缘多项式矩。本文详细阐述了该格式的构建过程，并通过导出不同范数下的误差估计证明了其收敛性。一系列代表性数值算例验证了该方法的表现。