We propose a goal-oriented mesh-adaptive algorithm for a finite element method stabilized via residual minimization on dual discontinuous-Galerkin norms. By solving a saddle-point problem, this residual minimization delivers a stable continuous approximation to the solution on each mesh instance and a residual projection onto a broken polynomial space, which is a robust error estimator to minimize the discrete energy norm via automatic mesh refinement. In this work, we propose and analyze a goal-oriented adaptive algorithm for this stable residual minimization. We solve the primal and adjoint problems considering the same saddle-point formulation and different right-hand sides. By solving a third stable problem, we obtain two efficient error estimates to guide goal-oriented adaptivity. We illustrate the performance of this goal-oriented adaptive strategy on advection–diffusion–reaction problems.

我们为有限元方法提出了一种面向目标的网格自适应算法，该方法通过对不连续Galerkin范数进行残差最小化来稳定。通过解决鞍点问题，此残差最小化可为每个网格实例上的解提供稳定的连续逼近，并为残破的多项式空间提供残差投影，这是可靠的误差估计器，可通过自动网格细化来最小化离散能量范数。在这项工作中，我们提出并分析了用于此稳定残差最小化的面向目标的自适应算法。考虑到相同的鞍点公式和不同的右侧，我们解决了原始和伴随问题。通过解决第三个稳定问题，我们获得了两个有效的误差估计值，以指导面向目标的适应性。