A Multigrid Method for the Pseudostress Formulation of Stokes Problems

Abstract

The purpose of this paper is to develop and analyze a multigrid solver for the finite element discretization of the pseudostress system associated with the differential operator $\mathcal{A}-\gamma\,\mathbf{grad}\,\mathbf{div}$ over $2\times 2$ matrix-valued functions. This system is derived from the pseudostress-velocity formulation [Z. Cai, B. Lee, and P. Wang, SIAM J. Numer. Anal., 42 (2004), pp. 843-859] of two-dimensional Stokes problems through the penalty method or natural time discretization for the respective steady- or unsteady-state problems. Here $\gamma>0$ is a constant associated with either the penalty parameter or the time-step size, and $\mathcal{A}$ is a singular map. In this paper, we develop a multigrid solver for the discrete problem using both the Raviart-Thomas (RT) and the Brezzi-Douglas-Marini (BDM) finite elements. We show that the multigrid convergence rate is $O(\frac{1+\gamma^{-2}}{1+\gamma^{-2} + m})$, where $m$ is the number of smoothings. This convergence rate is independent of the mesh size and the number of levels used in multigrid. Moreover, numerical results presented in this paper show that the multigrid convergence rate for the BDM elements does not depend on the parameter $\gamma$.