A Multigrid Method For Discrete Stokes Problems With Virtual Elements

In this paper,we study a multigrid method for the virtual element discretization of the Stokes problem based on non-nested polygonal meshes.Because the virtual element spaces on different levels of grids are non-nested,we propose a mesh transfer operator via the local projection,and prove the stability and approximation of the operator.Since discrete systems are indefinite,we define a pre-smooth and postsmooth operators respectively.We also prove that the W-cycle multigrid method is optimal under the condition of sufficiently large smoothing steps.Finally,some numerical tests are given.