Virtual element method for a nonlocal elliptic problem of Kirchhoff type on polygonal meshes

We consider the discretization of the nonlocal elliptic problem of Kirchhoff type using the virtual element method (VEM) over polygonal meshes. The nonlocal diffusion coefficient is approximated by using the L2 projection operator, which is directly computable from the degrees of freedom. However, the presence of the nonlocal term reduces the sparsity of the Jacobian matrix, which would increase the computational burden. To avoid this and to retain the sparsity of the Jacobian, the nonlinear system is replaced by an equivalent system inspired by the work on the FEM by Gudi (2012). The numerical results show that the proposed methodology yields optimal convergence rate in the L2 norm. Theoretical estimates are derived for H1 norms and are verified by solving a numerical example.