A numerical study of the virtual element method in anisotropic diffusion problems

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 fuctions 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.