Published

Au Auxiliary Space Multigrid Preconditioner for the Weak Galerkin Method

L. Chen, J. Wang, Y. Wang and X. Ye

Computers and Mathematics with Applications. 70(4), 330--344, 2015.

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ABSTRACT:

 In this paper, we construct an auxiliary space multigrid
preconditioner for the weak Galerkin method for second-order diffusion
equations, discretized on simplicial 2D or 3D meshes. The idea of the
auxiliary space multigrid preconditioner is to use an auxiliary space
as a "coarse" space in the multigrid algorithm, where the discrete
problem in the auxiliary space can be easily solved by an existing
solver. In our construction, we conveniently use the H1 conforming
piecewise linear finite element space as an auxiliary space. The main
technical difficulty is to build the connection between the weak
Galerkin discrete space and the H1 conforming piecewise linear finite
element space. We successfully constructed such an auxiliary space
multigrid preconditioner for the weak Galerkin method, as well as a
reduced system of the weak Galerkin method involving only the degrees
of freedom on edges/faces. The preconditioned systems are proved to
have condition numbers independent of the mesh size. Numerical
experiments further support the theoretical results.