In this study, we propose a virtual element scheme to solve the Darcy problem in three physical dimensions. The main novelty is that curved elements are naturally handled without any degradation of the solution accuracy. Indeed, in presence of curved boundaries, or internal interfaces, the geometrical error introduced by planar approximations may dominate the convergence rate limiting the benefit of high-order approximations. We consider the Darcy problem in its mixed form to directly obtain accurate and mass conservative fluxes without any post-processing. An important step to derive the proposed scheme is the integration over curved polyhedrons, here presented and discussed. Finally, we show the theoretical analysis of the scheme as well as several numerical examples to support our findings.

在这项研究中，我们提出了一种虚拟元素方案来解决三个物理维度的达西问题。主要的新颖之处在于可以自然地处理弯曲元素，而不会降低解决方案的准确性。实际上，在存在弯曲边界或内部界面的情况下，平面近似引入的几何误差可能会主导收敛速度，从而限制高阶近似的好处。我们考虑混合形式的达西问题，无需任何后处理即可直接获得准确的质量守恒通量。推导所提出方案的一个重要步骤是弯曲多面体的积分，这里介绍和讨论。最后，我们展示了该方案的理论分析以及几个数值示例来支持我们的发现。