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

MultiGrid Preconditioners for Mixed Finite Element Methods of Vector Laplacian