A Multigrid Algorithm For The Virtual Element Approximation Of The Second Order Elliptic Problems

In this paper,a multigrid algorithm is proposed for solving the discrete system of the virtual element method for the second order elliptic problem.In the first part of the paper,we introduce the virtual element method of the second-order elliptic problem.In the second part of the paper,because the virtual element spaces on different meshes are non nested,we first design the mesh transfer operator,give the multigrid algorithm,and then according to the framework in the literature[13],the convergence of W-cycle multigrid method is analyzed and proved,and the condition number estimation of the variable V-cycle multigrid preconditioning method is given.It is proved that the convergence of the W-cycle multigrid method and the condition number of the variable Vcycle multigrid preconditioning method is independent of the mesh size h of the quadrilateral mesh.Numerical examples demonstrate the effectiveness of the optimal convergence of the W-cycle multigrid method.