In this paper, we describe a stable finite element formulation for advection-diffusion-reaction problems that allows for robust automatic adaptive strategies to be easily implemented. We consider locally vanishing, heterogeneous, and anisotropic diffusivities, as well as advection-dominated diffusion problems. The general stabilized finite element framework was described and analyzed in arXiv:1907.12605v3 for linear problems in general, and tested for pure advection problems. The method seeks for the discrete solution through a residual minimization process on a proper stable discontinuous Galerkin (dG) dual norm. This technique leads to a saddle-point problem that delivers a stable discrete solution and a robust error estimate that can drive mesh adaptivity. In this work, we demonstrate the efficiency of the method in extreme scenarios, delivering stable solutions. The quality and performance of the solutions are comparable to classical discontinuous Galerkin formulations in the respective discrete space norm on each mesh. Meanwhile, this technique allows us to solve on coarse meshes and adapt the solution to achieve a user-specified solution quality.

中文翻译：

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通过残差最小化自动自适应稳定的有限元方法，用于求解非均质各向异性对流扩散反应问题

在本文中，我们描述了对流-扩散-反应问题的稳定有限元公式，使鲁棒的自动自适应策略易于实现。我们考虑局部消失，非均质和各向异性扩散，以及对流占主导的扩散问题。在arXiv：1907.12605v3中描述并分析了一般稳定的有限元框架，以解决一般线性问题，并测试了纯对流问题。该方法通过在适当的稳定不连续Galerkin（dG）对偶范数上的残差最小化过程寻求离散解。该技术导致鞍点问题，该问题提供了稳定的离散解决方案和鲁棒的误差估计，可以驱动网格自适应性。在这项工作中，我们演示了在极端情况下该方法的有效性，提供稳定的解决方案。该解决方案的质量和性能在每个网格上各自独立的空间范数上可与经典的不连续Galerkin公式相媲美。同时，该技术使我们能够求解粗网格并调整解决方案以实现用户指定的解决方案质量。