We describe a stable finite element formulation for advection–diffusion–reaction problems that allows for the easy implementation of robust automatic adaptivity. We consider locally vanishing, heterogeneous, and anisotropic diffusivities, as well as advection-dominated diffusion problems. We apply a general stabilized finite element framework (Calo et al., 2020) that seeks a discrete solution through a residual minimization process on a proper stable discontinuous Galerkin (dG) dual norm. This technique leads to a sequence of saddle-point problems that are discretely stable and deliver a robust error estimate that drives mesh adaptivity. In this work, we demonstrate the method’s efficiency in extreme scenarios, where the solutions’ quality and performance are comparable to classical discontinuous Galerkin formulations in the respective discrete space norm on a particular mesh. We focus on the practical implementation of the adaptive technology to a broad range of engineering applications, from singularly perturbed linear advection-dominated diffusion to highly non-linear problems. This technique starts from coarse meshes and adapts itself to achieve a user-specified solution quality.

我们为平流-扩散-反应问题描述了一个稳定的有限元公式，它允许轻松实现稳健的自动自适应。我们考虑局部消失、异质和各向异性扩散率，以及对流主导的扩散问题。我们应用了一个通用的稳定有限元框架 (Calo et al., 2020)，该框架通过适当稳定的残差最小化过程寻求离散解不连续伽辽金 (dG) 对偶范数。这种技术会导致一系列鞍点问题，这些问题是离散稳定的，并提供强大的误差估计，从而驱动网格自适应。在这项工作中，我们展示了该方法在极端场景中的效率，其中解决方案的质量和性能可与特定网格上各自离散空间范数中的经典不连续 Galerkin 公式相媲美。我们专注于自适应技术在广泛的工程应用中的实际应用，从奇异扰动的线性对流主导扩散到高度非线性问题。该技术从粗网格开始，并自行调整以实现用户指定的解决方案质量。