We prove the uniform convergence of the multigrid V-cycle on graded meshes for corner-like singularities of elliptic equations on a bounded two-dimensional domain P. In particular, using some weighted Sobolev space $K^m_a$ and the method of subspace corrections with the elliptic projection decomposition estimate on $K^m_a$, we show that the multigrid V-cycle converges uniformly for piecewise linear functions with standard smoothers (Richardson, weighted Jacobi, Gauss-Seidel, etc.). In addition, we have a similar argument on high-order polynomials and numerical results will be shown.