An improved multigrid algorithm for n-irregular meshes with subspace correction smoother

We propose a multigrid V-cycle algorithm for locally refined meshes with arbitrary hanging node configurations. Unlike existing algorithms that perform smoothing only on a subspace of the multigrid space, we adopt a global smoothing strategy at each multigrid level. This guarantees that an arbitrary improvement of the convergence bound can be obtained when increasing the number of smoothing iterations. When smoothing on a subspace, improvement can be obtained only up to a saturation value. The smoothing process we adopt is of successive subspace correction (SSC) type. The subspaces involved in the subspace decomposition of the multigrid space are chosen according to a multilevel strategy. This choice provides an easy way to deal with the hanging nodes generated by the local refinement procedure that allows the use of standard finite element codes. We present numerical results to highlight how the proposed algorithm has better convergence properties than local smoothing strategies that have a comparable computational complexity. Moreover, the numerical tests show that our method outperforms local smoothing approaches for Poisson's equation with discontinuous coefficients, where the solution is in H¹, but not in H².