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由线性目标泛函 $G$ 与一般二阶非对称线性椭圆型偏微分方程的解 $u^\star$ 所定义的目标量 $G(u^\star)$。当前非对称系统迭代代数求解器的分析缺乏泛函分析框架所规定的范数下的压缩性质，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，本文提出了一种目标导向自适应迭代对称化有限元法（GOAISFEM）。该方法采用嵌套循环结构，结合压缩对称化过程（例如Zarantonello迭代）与压缩代数求解器（例如最优多重网格求解器）。各种迭代过程需要精心设计的停止准则，以便自适应算法能有效引导局部网格细化与非精确离散逼近的计算。主要结果包括所提自适应算法的完全线性收敛性，以及关于自由度与总计算成本（即最优复杂度）的最优收敛率证明。数值实验验证了理论结果，并探讨了参数的选择。