An auxiliary space multigrid preconditioner for the weak Galerkin method

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 the 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.