The Virtual Element Method (VEM) is a novel family of numerical methods for approximating partial differential equations on very general polygonal or polyhedral computational grids. This work aims to propose a Balancing Domain Decomposition by Constraints (BDDC) preconditioner that allows using the conjugate gradient method to compute the solution of the saddle-point linear systems arising from the VEM discretization of the three-dimensional Stokes equations. We prove the scalability and quasi-optimality of the algorithm and confirm the theoretical findings with parallel computations. Numerical results with adaptively generated coarse spaces confirm the method's robustness in the presence of large jumps in the viscosity and with high-order VEM discretizations.

翻译：虚拟元方法（VEM）是一类新型数值方法，可在非常一般的多边形或多面体计算网格上逼近偏微分方程。本文旨在提出一种基于约束的平衡域分解（BDDC）预条件子，使得能够利用共轭梯度法求解由三维斯托克斯方程VEM离散化产生的鞍点线性系统。我们证明了该算法的可扩展性与准最优性，并通过并行计算验证了理论结果。基于自适应生成粗空间的数值实验表明，该方法在黏性系数存在大跳跃以及高阶VEM离散化情形下均具有鲁棒性。