Multigrid preconditioning of singularly perturbed convection-diffusion equations

TL;DR

It is shown that techniques from standard multigrid for anisotropic problems can be adapted to these discretizations on both tensor-product as well as semi-structured meshes, and are demonstrated to be robust preconditioners for several standard flow benchmarks.

Abstract

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.

Figures (6)

• Figure 1: Model problem solution to Equation \ref{['eq4']} with $\varepsilon=10^{-2}$. At left, the solution without SUPG shows significant oscillations in the exponential layer. At right, these wiggles are not present in the solution using SUPG.
• Figure 2: Left: Sample Shishkin mesh with $N=8$ for this dissection. Right: Dissection of $\Omega$ for unit-square domain based on transition points $\lambda_1$ and $\lambda_2$ and expected layer structure.
• Figure 3: Structured enumeration
• Figure 4: Unstructured enumeration
• Figure 6: The curvilinear mesh and solution at $\varepsilon=10^{-3}$

Theorems & Definitions (2)

• Example 3.1
• Example 3.2