  In this paper we study convergence estimates for a multigrid algorithm with
smoothers of successive subspace correction (SSC) type, applied to symmetric
elliptic PDEs. First, we revisit a general convergence analysis on a class of
multigrid algorithms in a fairly general setting, where no regularity
assumptions are made on the solution. In this framework, we are able to
explicitly highlight the dependence of the multigrid error bound on the number
of smoothing steps. For the case of no regularity assumptions, this represents
a new addition to the existing theory. Then, we analyze successive subspace
correction smoothing schemes for a set of uniform and local refinement
applications with either nested or non-nested overlapping subdomains. For these
applications, we explicitly derive bounds for the multigrid error, and identify
sufficient conditions for these bounds to be independent of the number of
multigrid levels. For the local refinement applications, finite element grids
with arbitrary hanging nodes configurations are considered. The analysis of
these smoothing schemes is cast within the far-reaching multiplicative Schwarz
framework.
