The use of orthonormal polynomial bases has been found to be efficient in preventing ill-conditioning of the system matrix in the primal formulation of Virtual Element Methods (VEM) for high values of polynomial degree and in presence of badly-shaped polygons. However, we show that using the natural extension of a orthogonal polynomial basis built for the primal formulation is not sufficient to cure ill-conditioning in the mixed case. Thus, in the present work, we introduce an orthogonal vector-polynomial basis which is built ad hoc for being used in the mixed formulation of VEM and which leads to very high-quality solution in each tested case. Furthermore, a numerical experiment related to simulations in Discrete Fracture Networks (DFN), which are often characterised by very badly-shaped elements, is proposed to validate our procedures.

翻译：在原始形式虚拟单元法（VEM）中，使用正交多项式基已被证明能有效防止高阶多项式次数及存在劣质多边形时系统矩阵的病态问题。然而，我们指出，直接将针对原始形式构建的正交多项式基进行自然扩展，并不足以解决混合形式中的病态问题。因此，在本工作中，我们引入了一种专为混合形式VEM设计的正交向量多项式基，该基在每种测试案例中均能获得极高质量的解。此外，我们通过一个与离散裂缝网络（DFN）相关的数值实验来验证所提方法，这类网络通常包含大量形状极差的单元。