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产生于流体流动的各类工程应用与设计可行性研究模型中。数十年来，其高效数值求解始终是一个备受关注且活跃的研究课题。在这些边值问题的有限元离散化背景下，流线迎风彼得罗夫-伽辽金（SUPG）技术能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，提出了高效的多重网格迭代求解方法。特别地，我们证明了各向异性问题的标准多重网格技术可适用于张量积网格及半结构化网格上的此类离散格式。实验表明，所提出的方法对多个标准流动基准测试具有鲁棒的预处理效果。