Goal-oriented error estimation provides the ability to approximate the discretization error in a chosen functional quantity of interest. Adaptive mesh methods provide the ability to control this discretization error to obtain accurate quantity of interest approximations while still remaining computationally feasible. Traditional discrete goal-oriented error estimates incur linearization errors in their derivation. In this paper, we investigate the role of linearization errors in adaptive goal-oriented error simulations. In particular, we develop a novel two-level goal-oriented error estimate that is free of linearization errors. Additionally, we highlight how linearization errors can facilitate the verification of the adjoint solution used in goal-oriented error estimation. We then verify the newly proposed error estimate by applying it to a model nonlinear problem for several quantities of interest and further highlight its asymptotic effectiveness as mesh sizes are reduced. In an adaptive mesh context, we then compare the newly proposed estimate to a more traditional two-level goal-oriented error estimate. We highlight that accounting for linearization errors in the error estimate can improve its effectiveness in certain situations and demonstrate that localizing linearization errors can lead to more optimal adapted meshes.

翻译：目标导向误差估计能够近似所选泛函量中的离散化误差。自适应网格方法可以控制这种离散化误差，从而在保持计算可行性的同时获得精确的感兴趣量近似。传统离散目标导向误差估计在其推导过程中引入了线性化误差。本文研究了线性化误差在自适应目标导向误差模拟中的作用。具体而言，我们提出了一种新型的、无线性化误差的两水平目标导向误差估计。此外，我们强调了线性化误差如何有助于验证目标导向误差估计中使用的伴随解。我们通过将该新误差估计应用于一个模型非线性问题的多个感兴趣量，验证了其有效性，并进一步突出了其随网格尺寸减小而渐近有效的特性。在自适应网格背景下，我们将新提出的估计与更传统的两水平目标导向误差估计进行了比较。结果表明，在误差估计中考虑线性化误差可以在某些情况下提高其有效性，并证明了局部化线性化误差能够生成更优的自适应网格。