  In this work, we propose three Braess-Sarazin-type multigrid relaxation
schemes for solving linear elasticity problems, where the marker and cell
scheme, a finite difference method, is used for the discretization. The three
relaxation schemes are Jacobi-Braess-Sarazin, Mass-Braess-Sarazin, and
Vanka-Braess-Sarazin. A local Fourier analysis (LFA) for the block-structured
relaxation schemes is presented to study multigrid convergence behavior. From
LFA, we derive optimal LFA smoothing factor for each case. We obtain highly
efficient smoothing factors, which are independent of Lam\'{e} constants.
Vanka-Braess-Sarazin relaxation scheme leads to the most efficient one. In each
relaxation, a Schur complement system needs to be solved. Due to the fact that
direct solve is often expensive, an inexact version is developed, where we
simply use at most three weighted Jacobi iterations on the Schur complement
system. Finally, two-grid and V-cycle multigrid performances are presented to
validate our theoretical results. Our numerical results show that inexact
versions can achieve the same performance as that of exact versions and our
methods are robust to the Lam\'{e} constants.
