The motive of the present work is to propose an adaptive numerical technique for singularly perturbed convection-diffusion problem in two dimensions. It has been observed that for small singular perturbation parameter, the problem under consideration displays sharp interior or boundary layers in the solution which cannot be captured by standard numerical techniques. In the present work, Hughes stabilization strategy along with the streamline upwind/Petrov-Galerkin (SUPG) method has been proposed to capture these boundary layers. Reliable a posteriori error estimates in energy norm on anisotropic meshes have been developed for the proposed scheme. But these estimates prove to be dependent on the singular perturbation parameter. Therefore, to overcome the difficulty of oscillations in the solution, an efficient adaptive mesh refinement algorithm has been proposed. Numerical experiments have been performed to test the efficiency of the proposed algorithm.

中文翻译：

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休斯稳定的SUPG技术的后验误差估计和对流扩散问题的自适应细化

本工作的目的是针对二维扰动对流扩散问题提出一种自适应数值技术。已经观察到，对于小的奇异摄动参数，所考虑的问题在解决方案中显示了尖锐的内部层或边界层，而这是标准数值技术无法捕获的。在目前的工作中，已经提出了休斯稳定策略以及流线上风/ Petrov-Galerkin（SUPG）方法来捕获这些边界层。对于所提出的方案，已经开发出各向异性网格上能量范数的可靠后验误差估计。但是这些估计证明依赖于奇异摄动参数。因此，为了克服溶液中振荡的困难，提出了一种高效的自适应网格细化算法。已经进行了数值实验以测试所提出算法的效率。