We consider the parallel solution of strongly anisotropic diffusion problems on adaptive grids by multigrid methods. The standard multigrid methods with pointwise relaxation and standard coarsening have problems because smoothers are not effective. To improve multigrid, the common ways are to use semicoarsening or line smoothers. However, neither of these techniques is suitable for parallel adaptive grids environment. Moreover, both techniques require that the anisotropy aligned with the grid, which may not be the case for realistic problems. In this talk, we will discuss a robust and easy-to-parallelize smoother for multigrid which can deal with strongly anisotropic problems on adaptive grids effectively. This method remains effective even if the anisotropy is not aligned with the grid.

We use multilevel adaptive technique (MLAT). The main idea of MLAT is to perform smoothing sweeps only on locally refined grids, and use the full approximation scheme (FAS) to generate the error correction cycle. We propose to use Krylov subspace methods as smoothers. Our numerical experiments show that they reduce oscillatory error components effectively. Therefore with such smoothers, multigrid achieves fast convergence rate. Moreover, parallelizing Krylov methods is straightforward since only matrix vector products and vector inner products need communication.

Convergence rate analysis for multigrid on adaptive grids can be simplified to analysis on uniform grids, since with properly chosen interpolation and prolongation operators, convergence rate on adaptive grids is almost identical to that on uniform grids. Standard analytic tools such as Local Fourier Analysis (LFA) fail for Krylov smoothers because they require the smoothing operators to be linear. In addition, Krylov methods are not ``strict'' smoothers. Their smoothing effect for high frequency modes may deteriorate when large smooth error components are present. In our work, we use a slightly different approach. Assume the coarse grid correction operator (including interpolation and restriction) satisfies certain requirements, we derive the level-independent upper bound for convergence rate of Krylov methods. This rate is used to estimate multigrid convergence rate. This explains why level-independent convergence rate of multigrid is achieved. The numerical experiments verify our statement. Our approach of quantitative analysis can be applied to more general problems and may be useful for other multigrid practitioners.

This work is part of IBEAM project which is sponsored under a Round III Grand Challenge Cooperative Agreement with NASA's Computational Technologies Project.