We present recent finite element numerical results on a model convection-diffusion problem in the singular perturbed case when the convection term dominates the problem. We compare the standard Galerkin discretization using the linear element with a saddle point least square discretization that uses quadratic test functions, trying to control and explain the non-physical oscillations of the discrete solutions. We also relate the up-winding Petrov-Galerkin method and the stream-line diffusion discretization method, by emphasizing the resulting linear systems and by comparing appropriate error norms. Some results can be extended to the multidimensional case in order to come up with efficient approximations for more general singular perturbed problems, including convection dominated models.

翻译：我们针对对流项占主导的奇异摄动情形下的模型对流扩散问题，展示了最新的有限元数值结果。通过采用二次检验函数的鞍点最小二乘离散化方法，我们将其与使用线性单元的标准伽辽金离散化进行对比，旨在控制并解释离散解中非物理振荡现象。通过强调最终生成的线性系统并比较适当的误差范数，我们进一步建立了迎风彼得罗夫-伽辽金方法与流线扩散离散化方法之间的关联。部分结果可推广至多维情形，从而为更一般的奇异摄动问题（包括对流主导模型）提出高效的近似方案。