  In this paper we study and compare two multigrid relaxation schemes with
coarsening by two, three, and four for solving elliptic sparse optimal control
problems with control constraints. First, we perform a detailed local Fourier
analysis (LFA) of a well-known collective Jacobi relaxation (CJR) scheme, where
the optimal smoothing factors are derived. This insightful analysis reveals
that the optimal relaxation parameters depend on mesh size and regularization
parameters, which was not investigated in literature. Second, we propose and
analyze a new mass-based Braess-Sarazin relaxation (BSR) scheme, which is
proven to provide smaller smoothing factors than the CJR scheme when $\alpha\ge
ch^4$ for a small constant $c$. Here $\alpha$ is the regularization parameter
and $h$ is the spatial mesh step size. These schemes are successfully extended
to control-constrained cases through the semi-smooth Newton method. Coarsening
by three or four with BSR is competitive with coarsening by two. Numerical
examples are presented to validate our theoretical outcomes. The proposed
inexact BSR (IBSR) scheme, where two preconditioned conjugate gradients
iterations are applied to the Schur complement system, yields better
computational efficiency than the CJR scheme.
