We analyse and study instability problems related to the solution of the advection–diffusion–reaction equation (ADR) using a standard finite element scheme. With this aim, this work has been carried out in the following way: first, three weak formulations are obtained from the general problem. In specific, we study the existence and uniqueness of the solution for each of the aforementioned formulations. Second, we analyse the general theory of consistent stabilisation techniques for the ADR equation, which includes: streamline upwind/Petrov-Galerkin (SUPG), and Galerkin/least-squares (GLS). Third, we study and develop, for linear triangular elements, two of the most important subgrid-scale techniques, i.e. algebraic subgrid scale (ASGS), and orthogonal subgrid scale (OSS). This includes the study of an expression for a stabilisation parameter based on an ADR equation's Fourier analysis. Finally, as conclusion, all these stabilisation techniques are put in context with the SUPG technique for a better comparison as well as understanding of their underlying features for linear triangular elements.

中文翻译：

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对流-扩散-反应方程有限元解一致稳定的经典技术分析

我们使用标准有限元方案分析和研究与对流-扩散-反应方程 (ADR) 的解相关的不稳定性问题。出于这个目的，这项工作已按以下方式进行：首先，从一般问题中获得三个弱公式。具体而言，我们研究了上述每个公式的解的存在性和唯一性。其次，我们分析了 ADR 方程的一致稳定技术的一般理论，其中包括：流线迎风/彼得罗夫-伽辽金 (SUPG) 和伽辽金/最小二乘法 (GLS)。第三，我们针对线性三角形单元研究和开发两种最重要的子网格尺度技术，即代数子网格尺度（ASGS）和正交子网格尺度（OSS）。这包括基于 ADR 方程的傅立叶分析研究稳定参数的表达式。最后，作为结论，所有这些稳定技术都与 SUPG 技术结合在一起，以便更好地比较和理解线性三角形元素的基本特征。