A Schwarz Preconditioner for a Hybridized Mixed Method

Abstract In this paper, we provide a Schwarz preconditioner for the hybridized versions of the Raviart-Thomas and Brezzi-Douglas-Marini mixed methods. The preconditioner is for the linear equation for Lagrange multipliers arrived at by eliminating the flux as well as the primal variable. We also prove a condition number estimate for this equation when no preconditioner is used. Although preconditioners for the lowest-order case of the Raviart-Thomas method have been constructed previously by exploiting its connection with a nonconforming method, our approach is different in that we use a new variational characterization of the Lagrange multiplier equation. This allows us to precondition even the higher-order cases of these methods.

**摘要**：在本文中，我们为Raviart - Thomas和Brezzi - Douglas - Marini混合方法的杂交形式提供了一个Schwarz预条件子。该预条件子是针对通过消去通量以及原始变量而得到的关于拉格朗日乘子的线性方程。我们还在未使用预条件子时证明了该方程的条件数估计。尽管先前通过利用Raviart - Thomas方法最低阶情形与非协调方法的联系构建了其预条件子，但我们的方法不同，我们使用了拉格朗日乘子方程的一种新的变分特征。这使我们甚至能够对这些方法的高阶情形进行预条件处理。