We consider a linear symmetric and elliptic PDE and a linear goal functional. We design and analyze a goal-oriented adaptive finite element method, which steers the adaptive mesh-refinement as well as the approximate solution of the arising linear systems by means of a contractive iterative solver like the optimally preconditioned conjugate gradient method or geometric multigrid. We prove linear convergence of the proposed adaptive algorithm with optimal algebraic rates. Unlike prior work, we do not only consider rates with respect to the number of degrees of freedom but even prove optimal complexity, i.e., optimal convergence rates with respect to the total computational cost.

翻译：我们认为一个线性对称和椭圆PDE以及一个线性目标功能。我们设计并分析一个面向目标的适应性有限元素方法,该方法通过一个合同式迭代求解器,如最佳先决条件的同源梯度法或几何多格,指导适应性网状精炼以及新产生的线性系统的近似解决办法。我们证明,拟议的适应性算法与最佳代数率呈线性趋同。与以往的工作不同,我们不仅考虑自由度的速率,而且甚至证明是最佳复杂性,即总计算成本的最佳趋同率。