The virtual element method is well suited to the formulation of arbitrarily regular Galerkin approximations of elliptic partial differential equations of order 2p1, for any integer p1 ≥ 1. In fact, the virtual element paradigm provides a very effective design framework for conforming, finite dimensional subspaces of Hp2(Ω), Ω being the computational domain and p2 ≥ p1 another suitable integer number. In this review, we first present an abstract setting for such highly regular approximations and discuss the mathematical details of how we can build conforming approximation spaces with a global high-order regularity on Ω. Then, we illustrate specific examples in the case of second- and fourth-order partial differential equations, that correspond to the cases p1 = 1 and 2, respectively. Finally, we investigate numerically the effect on the approximation properties of the conforming highly-regular method that results from different choices of the degree of continuity of the underlying virtual element spaces and how different stabilization strategies may impact on convergence.

中文翻译：

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二阶及高阶椭圆偏微分方程的任意正则一致虚元方法综述

虚元法非常适用于椭圆偏微分方程的任意正则 Galerkin 近似的公式化2p1, 对于任何整数p1 ≥ 1. 事实上，虚拟元素范式为符合的有限维子空间提供了一个非常有效的设计框架。Hp2(Ω),Ω是计算域和p2 ≥ p1另一个合适的整数。在这篇综述中，我们首先提出了这种高度正则近似的抽象设置，并讨论了如何在Ω. 然后，我们举例说明二阶和四阶偏微分方程的具体例子，它们对应的情况p1 = 1和2， 分别。最后，我们在数值上研究了对一致的高度正则方法的逼近特性的影响，这种影响是由于底层虚拟元素空间的连续性程度的不同选择以及不同的稳定策略如何影响收敛性而产生的。