For a model convection-diffusion problem, we obtain new error estimates for a general upwinding finite element discretization based on bubble modification of the test space. The key analysis tool is based on finding representations of the optimal norms on the trial spaces at the continuous and discrete levels. We analyze and compare the standard linear discretization, the saddle point least square and upwinding Petrov-Galerkin methods. We conclude that the bubble upwinding Petrov-Galerkin method is the most performant discretization for the one dimensional model. Our results for the model convection-diffusion problem can be extended for creating new and efficient discretizations for the multidimensional cases.

翻译：针对模型对流-扩散问题，我们基于测试空间的泡函数修正，获得了一种通用迎风有限元离散化的新误差估计。关键分析工具在于寻找连续和离散水平上试验空间最优范数的表示。我们分析并比较了标准线性离散化、鞍点最小二乘法和迎风Petrov-Galerkin方法。结论表明，对于一维模型，泡函数迎风Petrov-Galerkin方法是性能最优的离散化方案。本模型对流-扩散问题的研究成果可推广至多维情形，为构建新型高效离散化方法提供基础。