Abstract In this paper, we present the Virtual Element Method (VEM) for the solution of strongly anisotropic diffusion equations. In the VEM, the bilinear form associated with the diffusion equations is decomposed into two parts: a consistency term and a stability term. Therefore, the local stiffness matrix is the sum of two matrices: a consistency matrix and a stability matrix. Both matrices are constructed by using suitable projection operators that are computable from the degrees of freedom. The VEM stiffness matrix becomes very ill-conditioned in presence of a strong anisotropy of the diffusion tensor coefficient, leading to a loss of convergence, an effect known in the literature as mesh locking. In this work, we compare different choices of the stabilization, the basis functions and the elliptic projection operator, in order to alleviate the mesh locking phenomenon. To this end, we use orthonormal basis functions for the space of polynomials of degree k and an elliptic projection operator that is weighted with respect to the diffusion tensor. Moreover, the VEM with k = 1 needs a particular treatment to avoid locking. Numerical experiments with different values of k confirm the validity of the proposed approach.

中文翻译：

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各向异性扩散问题中虚元法的数值研究

摘要 在本文中，我们提出了求解强各向异性扩散方程的虚拟元法（VEM）。在 VEM 中，与扩散方程相关的双线性形式被分解为两部分：一致性项和稳定性项。因此，局部刚度矩阵是两个矩阵的总和：一致性矩阵和稳定性矩阵。两个矩阵都是通过使用可根据自由度计算的合适的投影算子构建的。VEM 刚度矩阵在扩散张量系数存在强各向异性的情况下变得非常病态，导致收敛损失，这种效应在文献中称为网格锁定。在这项工作中，我们比较了稳定性、基函数和椭圆投影算子的不同选择，以缓解锁网现象。为此，我们对 k 次多项式空间使用正交基函数，并使用相对于扩散张量加权的椭圆投影算子。此外，k = 1 的 VEM 需要特殊处理以避免锁定。具有不同 k 值的数值实验证实了所提出方法的有效性。