论文标题
用于stokes的完全无差异的不连续的Galerkin方法的代数预处理
An algebraic preconditioner for the exactly divergence-free discontinuous Galerkin method for Stokes
论文作者
论文摘要
我们为Cockburn，Kanschat和Schötzau提供了完全无差异的不连续的Galerkin（DG）离散化[J.科学。 Comput。，31（2007），第61--73页和王和Ye [Siam J. Numer。肛门，45（2007），第1269---1286页，有关Stokes问题。该DG方法使用使用$ h（{\ rm div}）$的有限元素 - 符合基础，从而通过迭代方法显着使其解决方案复杂化。已经开发了一些用于此Stokes离散化的预处理，但每个都基于专门的求解器或分解，并且没有提供明确的框架来推广到Navier-Stokes。为了避免需要定制的求解器，我们将$ h（{\ rm div}）$杂交 - 合格有限元，以使速度生活在标准的$ l^2 $ -DG空间中，并为扩展的杂交系统提供一个简单的代数预处理。所提出的预处理在$ h $中是最佳的，在元素顺序上超级稳健（最多显示第5阶），在2D和3D方面有效，并且仅依赖于标准的放松和代数的多机方法。杂交也自然地扩展到Navier-Stokes，为有效的黑盒预防剂提供了潜在的途径，以确切地对Navier-Stokes进行无差异的DG离散。
We present an optimal preconditioner for the exactly divergence-free discontinuous Galerkin (DG) discretization of Cockburn, Kanschat, and Schötzau [J. Sci. Comput., 31 (2007), pp. 61--73] and Wang and Ye [SIAM J. Numer. Anal., 45 (2007), pp. 1269--1286] for the Stokes problem. This DG method uses finite elements that use an $H({\rm div})$-conforming basis, thereby significantly complicating its solution by iterative methods. Several preconditioners for this Stokes discretization have been developed, but each is based on specialized solvers or decompositions, and do not offer a clear framework to generalize to Navier--Stokes. To avoid requiring custom solvers, we hybridize the $H({\rm div})$-conforming finite element so that the velocity lives in a standard $L^2$-DG space, and present a simple algebraic preconditioner for the extended hybridized system. The proposed preconditioner is optimal in $h$, super robust in element order (demonstrated up to 5th order), effective in 2d and 3d, and only relies on standard relaxation and algebraic multigrid methods available in many packages. The hybridization also naturally extends to Navier--Stokes, providing a potential pathway to effective black-box preconditioners for exactly divergence-free DG discretizations of Navier--Stokes.
