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Geometric versus algebraic multigrid
Standard AMG coarsening
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5-point accelerated aggregates aggressive coarsening algebraic multigrid algebraically smooth error AMG's anisotropies applied approximation assume C-variables C/F-splitting coarse-level coarsening strategy computing conjugate gradient conjugate gradient method Consequently considered convergence factor Convergence histories corresponding denotes direct solver Dirichlet boundary conditions discretization efficient eigenvalues eigenvectors equations example F-relaxation F-smoothing faster Figure Galerkin operator Gauss-Seidel relaxation geometric multigrid grid instance interpolatory isotropic iteration Jacobi relaxation Jacobi-interpolation Kh,H Lemma linear interpolation M-matrices matrices memory requirement multigrid methods n-connected negative norm obtain operator complexity particular performance piecewise constant piecewise constant interpolation Poisson-like positive connections positive definite post-smoothing pre-conditioner problems reduced relaxation steps Remark resulting robust satisfied Schur complement setup smoothing property solution solve stand-alone standard coarsening standard interpolation stencils strong connectivity strongly diagonally dominant strongly n-coupled substantially Theorem two-level convergence typical uniform convergence V-cycle convergence variables variants variational principle vector VS(S VS(S)-cycle weakly diagonally dominant