This work considers the iterative solution of large-scale problems subject to non-symmetric matrices or operators arising in discretizations of (port-)Hamiltonian partial differential equations. We consider problems governed by an operator $\mathcal{A}=\mathcal{H}+\mathcal{S}$ with symmetric part $\mathcal{H}$ that is positive (semi-)definite and skew-symmetric part $\mathcal{S}$. Prior work has shown that the structure and sparsity of the associated linear system enables Krylov subspace solvers such as the generalized minimal residual method (GMRES) or short recurrence variants such as Widlund's or Rapoport's method using the symmetric part $\mathcal{H}$, or an approximation of it, as preconditioner. In this work, we analyze the resulting condition numbers, which are crucial for fast convergence of these methods, for various partial differential equations (PDEs) arising in diffusion phenomena, fluid dynamics, and elasticity. We show that preconditioning with the symmetric part leads to a condition number uniform in the mesh size in case of elliptic and parabolic PDEs where $\mathcal{H}^{-1}\mathcal{S}$ is a bounded operator. Further, we employ the tailored Krylov subspace methods in optimal control by means of a condensing approach and a constraint preconditioner for the optimality system. We illustrate the results by various large-scale numerical examples and discuss efficient evaluations of the preconditioner, such as incomplete Cholesky factorization or the algebraic multigrid method.

翻译：本文研究在(端口-)哈密顿偏微分方程离散化过程中产生的非对称矩阵或算子所对应的大规模问题的迭代求解。我们考虑由算子$\mathcal{A}=\mathcal{H}+\mathcal{S}$控制的问题，其中对称部分$\mathcal{H}$是正定（半正定）的，而斜对称部分为$\mathcal{S}$。先前的研究表明，相关线性系统的结构与稀疏性使得Krylov子空间求解器（如广义最小残差法GMRES）或短递推变体（如Widlund方法或Rapoport方法）能够以对称部分$\mathcal{H}$或其近似作为预条件子。本文针对扩散现象、流体动力学和弹性力学中出现的各类偏微分方程，分析了这些方法快速收敛的关键因素——所得条件数。我们证明，在椭圆型和抛物型PDE情形下（其中$\mathcal{H}^{-1}\mathcal{S}$为有界算子），采用对称部分进行预条件处理可获得与网格尺寸无关的一致条件数。进一步地，我们通过凝聚化方法和最优性系统的约束预条件子，将定制的Krylov子空间方法应用于最优控制问题。最后通过多个大规模数值算例展示结果，并讨论预条件子的高效实现方案（如不完全Cholesky分解或代数多重网格方法）。