We present goal-oriented a posteriori error estimates for the automatic variationally stable finite element (AVS-FE) method (Calo et al., 2020) for scalar-valued convection–diffusion problems. The AVS-FE method is a Petrov–Galerkin method in which the test space is broken, whereas the trial space consists of classical FE basis functions, e.g., $C^0$ or Raviart–Thomas functions. We employ the concept of optimal test functions of the discontinuous Petrov–Galerkin (DPG) method by Demkowicz and Gopalakrishnan (Demkowicz and Gopalakrishnan, 2010; Carstensen et al., 2014; Demkowicz and Gopalakrishnan, 2011a; Demkowicz and Gopalakrishnan, 2011b; Demkowicz and Gopalakrishnan, 2012), leading to unconditionally stable FE approximations. Remarkably, by using $C^0$ or Raviart–Thomas trial spaces, the optimal discontinuous test functions can be computed in a completely decoupled element-by-element fashion.

To establish the error estimators we present two approaches: (i) following the classical approach of Becker and Rannacher (Becker and Rannacher, 2001), i.e., the dual solution is sought in the (broken) test space, and (ii) introducing an alternative approach in which we seek $C^0$, or Raviart–Thomas, AVS-FE approximations of the dual solution by using the underlying strong form of the dual boundary value problem (BVP). Various numerical verifications for 2D convection-dominated diffusion BVPs show that the estimates of the approximation error by the new alternative method are highly accurate, while the classical approach leads to error estimates of poor quality. Lastly, we present an algorithm for $h$-adaptive processes based on control of the numerical approximation error via the new alternative approach. Numerical verifications show that the estimator maintains high accuracy as the error converges to zero.

对于标量值对流扩散问题，我们针对自动变分稳定有限元（AVS-FE）方法（Calo等人，2020年）提出了面向目标的后验误差估计。AVS-FE方法是一种Petrov-Galerkin方法，其中测试空间被破坏，而试用空间则由经典的FE基础函数组成，例如，$C^0$或Raviart–Thomas功能。我们采用Demkowicz和Gopalakrishnan（Demkowicz和Gopalakrishnan，2010; Carstensen等人，2014; Demkowicz和Gopalakrishnan，2011a; Demkowicz和Gopalakrishwinan，2011b; Demkowicz和Gopalakrishwinan，2011b; Demkowicz和Gopalakrishwinan，2011b; Demkowicz和Gopalakrishwinan，2011b; Demkowicz和Gopalakrishnan，2011b; Demkowicz和Gopalakrishwinan，2011b; Demkowicz和Gopalakrishwinan，2011b; Demkowicz和Gopalakrishwinan，2011b; Demkowicz和Gopalakrishwinan，2011b; Gopalakrishnan，2012年），导致无条件稳定的有限元逼近。显着地，通过使用$C^0$ 或Raviart–Thomas试用空间，可以以完全分离的逐元素方式计算最佳不连续测试函数。

为了建立误差估计量，我们提出了两种方法：（i）遵循贝克尔和兰纳赫的经典方法（贝克尔和兰纳赫，2001年），即在（破碎的）测试空间中寻求对偶解，以及（ii）引入我们寻求的替代方法$C^0$或Raviart–Thomas，通过使用双重边值问题（BVP）的潜在强形式对偶解的AVS-FE逼近。对二维对流占主导的扩散BVP的各种数值验证表明，通过新的替代方法估算的近似误差非常准确，而经典方法则导致质量较差的误差估算。最后，我们提出一种算法$H$通过新的替代方法控制数值逼近误差的自适应过程。数值验证表明，当误差收敛到零时，估计器可保持较高的精度。