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For the curl curl test, if beta = 1e-9 alpha/dx^2, the multigrid solver is able to reduce the residual by 10 orders of magnitude in 10 v-cycles. But for beta = 1e-14 alpha/dx^2, the multigrid solver's residual will stall at about 5e-6 of the original residual. However, it can be solved using GMRES with multigrid as preconditioner.