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In this paper we study and compare two multigrid relaxation schemes withcoarsening by two, three, and four for solving elliptic sparse optimal controlproblems with control constraints. First, we perform a detailed local Fourieranalysis (LFA) of a well-known collective Jacobi relaxation (CJR) scheme, wherethe optimal smoothing factors are derived. This insightful analysis revealsthat the optimal relaxation parameters depend on mesh size and regularizationparameters, which was not investigated in literature. Second, we propose andanalyze a new mass-based Braess-Sarazin relaxation (BSR) scheme, which isproven to provide smaller smoothing factors than the CJR scheme when $\alpha\gech^4$ for a small constant $c$. Here $\alpha$ is the regularization parameterand $h$ is the spatial mesh step size. These schemes are successfully extendedto control-constrained cases through the semi-smooth Newton method. Coarseningby three or four with BSR is competitive with coarsening by two. Numericalexamples are presented to validate our theoretical outcomes. The proposedinexact BSR (IBSR) scheme, where two preconditioned conjugate gradientsiterations are applied to the Schur complement system, yields bettercomputational efficiency than the CJR scheme.

Smoothing analysis of two robust multigrid methods for elliptic optimal control problems