Bounds on the spectrum of Schur complements of subdomain stiffness matrices with respect to the interior variables are key ingredients of the convergence analysis of FETI (finite element tearing and interconnecting) based domain decomposition methods. Here we give bounds on the regular condition number of Schur complements of ‘floating’ clusters arising from the discretization of 3D Laplacian on a cube decomposed into cube subdomains. The results show that the condition number of the cluster defined on a fixed domain decomposed into m × m × m cube subdomains connected by face and optionally edge averages increases proportionally to m . The estimates support scalability of unpreconditioned H-FETI-DP (hybrid FETI dual-primal) method. Though the research is most important for the solution of variational inequalities, the results of numerical experiments indicate that unpreconditioned H-FETI-DP with large clusters can be useful also for the solution of huge linear problems.

中文翻译：

---

Schur 补充大型混合 FETI-DP 簇和巨大的三维标量问题的谱界

子域刚度矩阵的 Schur 补谱相对于内部变量的边界是基于 FETI（有限元撕裂和互连）的域分解方法的收敛性分析的关键要素。在这里，我们给出了由 3D 拉普拉斯算子在分解为立方体子域的立方体上的离散化产生的“浮动”簇的 Schur 补集的规则条件数的界限。结果表明，在固定域上定义的簇的条件数分解为 m × m × m 由面和可选边缘平均值连接的立方体子域，与 m 成比例增加。估计支持未预处理的 H-FETI-DP（混合 FETI 双原始）方法的可扩展性。虽然这项研究对于解决变分不等式是最重要的，