This paper interprets the stabilized finite element method via residual minimization as a variational multiscale method. We approximate the solution to the partial differential equations using two discrete spaces that we build on a triangulation of the domain; we denote these spaces as coarse and enriched spaces. Building on the adaptive stabilized finite element method via residual minimization, we find a coarse-scale approximation in a continuous space by minimizing the residual on a dual discontinuous Galerkin norm; this process allows us to compute a robust error estimate to construct an on-the-fly adaptive method. We reinterpret the residual projection using the variational multiscale framework to derive a fine-scale approximation. As a result, on each mesh of the adaptive process, we obtain stable coarse- and fine-scale solutions derived from a symmetric saddle-point formulation and an a-posteriori error indicator to guide automatic adaptivity. We test our framework in several challenging scenarios for linear and nonlinear convection-dominated diffusion problems to demonstrate the framework's performance in providing stability in the solution with optimal convergence rates in the asymptotic regime and robust performance in the pre-asymptotic regime. Lastly, we introduce a heuristic dual-term contribution in the variational form to improve the full-scale approximation for symmetric formulations (e.g., diffusion problem).

翻译：本文将基于残差最小化的稳定化有限元方法解释为一种变分多尺度方法。我们利用在域三角剖分上构建的两个离散空间来逼近偏微分方程的解，并将这些空间分别称为粗空间和富集空间。基于通过残差最小化建立的自适应稳定化有限元方法，我们通过在对偶间断伽辽金范数上最小化残差，在连续空间中寻找粗尺度逼近；该过程能够计算鲁棒的误差估计，从而构建即时自适应方法。我们利用变分多尺度框架重新解释残差投影，以推导细尺度逼近。由此，在自适应过程的每个网格上，我们获得由对称鞍点公式导出的稳定粗尺度与细尺度解，并得到用于引导自动自适应性的后验误差指示器。我们在线性和非线性对流主导扩散问题的多个具有挑战性的场景中测试框架，以证明其在渐近区域提供最优收敛速度的稳定性以及在预渐近区域的鲁棒性能。最后，我们在变分形式中引入启发式对偶项，以改进对称公式（例如扩散问题）的全尺度逼近。