Analysis and Comparison of Geometric and Algebraic Multigrid for Convection‐Diffusion Equations

Abstract

The discrete convection-diffusion equations obtained from streamline diffusion finite element discretization are solved on both uniform meshes and adaptive meshes. Estimates of error reduction rates for both geometric multigrid (GMG) and algebraic multigrid (AMG) are established on uniform rectangular meshes for a model problem. Our analysis shows that GMG with line Gauss-Seidel smoothing and bilinear interpolation converges if $h\gg \epsilon^{2/3}$, and AMG with the same smoother converges more rapidly than GMG if the interpolation constant $\beta$ in the approximation assumption of AMG satisfies $\beta \ll (\frac{h}{\sqrt{\epsilon}})^{\alpha}, \hs{1mm} \mbox{where $ \alpha = \Big\{ {\begin{smallmatrix} 1, & {h < \sqrt \varepsilon, } 2, & {h \ge \sqrt \varepsilon.} \end{smallmatrix}}$}$ On unstructured triangular meshes, the performance of GMG and AMG, both as solvers and as preconditioners for GMRES, are evaluated. Numerical results show that GMRES with AMG preconditioning is a robust and reliable solver on both type of meshes.