Published

Multi-Grid Preconditioners for Mixed Finite Element Methods of Vector Laplacian

Long Chen, Yongke Wu, Lin Zhong, Jie Zhou

Journal of Scientific Computing

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ABSTRACT:

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.