AI 中文总结
本文对界面控制域分解（ICDD）方法进行代数收敛性分析，结合谱估计与GMRES理论，证明收敛速率与重叠宽度和多项式次数相关而与网格尺寸无关，并推广至SRAS和RAS方法。
AI 中文摘要
我们发展了界面控制域分解（ICDD）方法的收敛性分析，该方法是一种基于最优控制框架的重叠域分解方法，采用Dirichlet界面控制函数和界面观测。我们考虑由每个子域中的$hp-$有限元法逼近的可能具有不连续系数的椭圆问题。当离散化在二维域之间的重叠区域上是一致的（conforming）时，我们提供了求解与ICDD相关的非对称界面Schur补系统所需的GMRES迭代次数的理论估计。我们的结果通过结合ICDD的Schur补矩阵的新颖谱估计和经典GMRES收敛理论获得。我们证明了收敛速率表现为$\mathcal{O}(\delta^{-1} p^{3/2}\log p)$，其中$\delta$表示重叠宽度，$p$表示局部多项式次数，同时与网格尺寸$h$无关。数值实验验证了理论预测，并展示了ICDD在计算域中存在大系数跳跃时的有效性。由于在一致（conforming）情况下，所考虑的ICDD公式与子结构限制加性Schwarz（SRAS）方法一致，该分析也为二维SRAS方法提供了收敛性估计，并通过其已知的等价性，为限制加性Schwarz（RAS）方法提供了收敛性估计。
英文摘要
We develop the convergence analysis of the Interface Control Domain Decomposition (ICDD) method, an overlapping domain decomposition method based on an optimal control framework with Dirichlet interface control functions and interface observation. We consider elliptic problems with possible discontinuous coefficients approximated by $hp-$FEM in each subdomain. When the discretizations are conforming on the overlap between 2D domains, we provide theoretical estimates of the number of GMRES iterations needed to solve the non-symmetric interface Schur complement system associated with ICDD. Our results are obtained by combining novel spectral estimates for the Schur complement matrix of ICDD and classical GMRES convergence theory. We prove that the convergence rate behaves as $\mathcal{O}(δ^{-1} p^{3/2}\log p)$, where $δ$ denotes the overlap width and $p$ the local polynomial degree, while remaining independent of the mesh size $h$. Numerical experiments verify the theoretical predictions and show the effectiveness of ICDD in the presence of large coefficient jumps in the computational domain. Since in the conforming case, the considered ICDD formulation coincides with the Substructured Restricted Additive Schwarz (SRAS) method, the analysis also provides convergence estimates for SRAS in two dimensions and, through its known equivalence, for the Restricted Additive Schwarz (RAS) method.
