  Due to the indefiniteness and poor spectral properties, the discretized
linear algebraic system of the vector Laplacian by mixed finite element methods
is hard to solve. A block diagonal preconditioner has been developed and shown
to be an effective preconditioner by Arnold, Falk, and Winther [Acta Numerica,
15:1--155, 2006]. The purpose of this paper is to propose alternative and
effective block diagonal and block triangular preconditioners for solving this
saddle point system. A variable V-cycle multigrid method with the standard
point-wise Gauss-Seidel smoother is proved to be a good preconditioner for a
discrete vector Laplacian operator. This multigrid solver will be further used
to build preconditioners for the saddle point systems of the vector Laplacian
and the Maxwell equations with divergent free constraint. The major benefit of
our approach is that the point-wise Gauss-Seidel smoother is more algebraic and
can be easily implemented as a black-box smoother.
