Boundary value problems based on the convection-diffusion equation arise naturally in models of fluid flow across a variety of engineering applications and design feasibility studies. Naturally, their efficient numerical solution has continued to be an interesting and active topic of research for decades. In the context of finite-element discretization of these boundary value problems, the Streamline Upwind Petrov-Galerkin (SUPG) technique yields accurate discretization in the singularly perturbed regime. In this paper, we propose efficient multigrid iterative solution methods for the resulting linear systems. In particular, we show that techniques from standard multigrid for anisotropic problems can be adapted to these discretizations on both tensor-product as well as semi-structured meshes. The resulting methods are demonstrated to be robust preconditioners for several standard flow benchmarks.

翻译：基于对流扩散方程的边值问题自然出现在多种工程应用及设计可行性研究的流体流动模型中。因此，其高效数值解数十年来一直是有趣且活跃的研究课题。在对这些边值问题进行有限元离散的背景下，流线上迎风Petrov-Galerkin (SUPG) 技术能够在奇异摄动区域实现精确离散。本文针对由此产生的线性系统，提出高效的多重网格迭代求解方法。特别地，我们证明各向异性问题标准多重网格技术可适用于张量积网格和半结构化网格上的此类离散。结果表明，所提出方法在多个标准流动基准测试中均表现出稳健的预处理性能。