We analyze a goal-oriented adaptive algorithm that aims to efficiently compute the quantity of interest $G(u^\star)$ with a linear goal functional $G$ and the solution $u^\star$ to a general second-order nonsymmetric linear elliptic partial differential equation. The current state of the analysis of iterative algebraic solvers for nonsymmetric systems lacks the contraction property in the norms that are prescribed by the functional analytic setting. This seemingly prevents their application in the optimality analysis of goal-oriented adaptivity. As a remedy, this paper proposes a goal-oriented adaptive iteratively symmetrized finite element method (GOAISFEM). It employs a nested loop with a contractive symmetrization procedure, e.g., the Zarantonello iteration, and a contractive algebraic solver, e.g., an optimal multigrid solver. The various iterative procedures require well-designed stopping criteria such that the adaptive algorithm can effectively steer the local mesh refinement and the computation of the inexact discrete approximations. The main results consist of full linear convergence of the proposed adaptive algorithm and the proof of optimal convergence rates with respect to both degrees of freedom and total computational cost (i.e., optimal complexity). Numerical experiments confirm the theoretical results and investigate the selection of the parameters.

翻译：本文分析了一种目标导向自适应算法，该算法旨在高效计算目标量$G(u^\star)$，其中$G$为线性目标泛函，$u^\star$为一般二阶非对称线性椭圆型偏微分方程的解。当前针对非对称系统的迭代代数求解器分析，在泛函分析框架所规定的范数中缺乏压缩性。这似乎阻碍了其在目标导向自适应最优性分析中的应用。为此，本文提出一种目标导向自适应迭代对称化有限元方法（GOAISFEM）。该方法采用嵌套循环结构，包含一个具有压缩性的对称化过程（如Zarantonello迭代）和一个具有压缩性的代数求解器（如最优多重网格求解器）。各迭代过程需要精心设计停止准则，使得自适应算法能够有效指导局部网格细化以及非精确离散近似的计算。主要结果包括：所提自适应算法具有完全线性收敛性，并在自由度与总计算成本两方面均证明了最优收敛速率（即最优复杂度）。数值实验验证了理论结果，并对参数选择进行了探究。