Uniform convergence of the multigrid V -cycle on graded meshes for corner singularities

This paper analyzes a multigrid (MG) V-cycle scheme for solving the discretized 2D Poisson equation with corner singularities. Using weighted Sobolev spaces K^m_a(Ω) and a space decomposition based on elliptic projections, we prove that the MG V-cycle with standard smoothers (Richardson, weighted Jacobi, Gauss- Seidel, etc.) and piecewise linear interpolation converges uniformly for the linear systems obtained by finite element discretization of the Poisson equation on graded meshes. In addition, we provide numerical experiments to demonstrate the optimality of the proposed approach.