论文标题
BDDC预处理用于差异的divergence the stokes方程的虚拟元素离散
BDDC preconditioners for divergence free virtual element discretizations of the Stokes equations
论文作者
论文摘要
虚拟元素方法（VEM）是用于近似偏微分方程的数值方法的新系列，其中多面网格元素的几何形状可能非常笼统。本文的目的是通过约束（BDDC）预调节器将平衡域分解扩展到由二维Stokes方程的VEM离散化产生的鞍点线性系统的解决方案。在原始未知数选择的合适下降下，预处理的线性系统结果对称和正确定，因此可以将预处理的共轭梯度方法用于其解决方案。我们提供了理论收敛分析，估计预处理线性系统的状况数量。几个数值实验验证了理论估计值，显示了提出的方法的可伸缩性和准理想性。此外，求解器相对于多边形元素的形状表现出强大的行为。我们还表明，可以通过易于实现的粗糙空间来实现更快的融合，比该理论所涵盖的融合略大。
The Virtual Element Method (VEM) is a new family of numerical methods for the approximation of partial differential equations, where the geometry of the polytopal mesh elements can be very general. The aim of this article is to extend the balancing domain decomposition by constraints (BDDC) preconditioner to the solution of the saddle-point linear system arising from a VEM discretization of the two-dimensional Stokes equations. Under suitable hypotesis on the choice of the primal unknowns, the preconditioned linear system results symmetric and positive definite, thus the preconditioned conjugate gradient method can be used for its solution. We provide a theoretical convergence analysis estimating the condition number of the preconditioned linear system. Several numerical experiments validate the theoretical estimates, showing the scalability and quasi-optimality of the method proposed. Moreover, the solver exhibits a robust behavior with respect to the shape of the polygonal mesh elements. We also show that a faster convergence could be achieved with an easy to implement coarse space, slightly larger than the minimal one covered by the theory.
