Optimal-order nonnested multigrid methods for solving finite element equations. I. On quasi-uniform meshes

Abstract:

We prove that the multigrid method works with optimal computational order even when the multiple meshes are not nested. When a coarse mesh is not a submesh of the finer one, the coarse-level correction usually does not have the $a( \cdot , \cdot )$ projection property and does amplify the iterative error in some components. Nevertheless, the low-frequency components of the error can still be caught by the coarse-level correction. Since the (amplified) high-frequency errors will be damped out by the fine-level smoothing efficiently, the optimal work order of the standard multigrid method can still be maintained. However, unlike the case of nested meshes, a nonnested multigrid method with one smoothing does not converge in general, no matter whether it is a V-cycle or a W-cycle method. It is shown numerically that the convergence rates of nonnested multigrid methods are not necessarily worse than those of nested ones. Since nonnested multigrid methods accept quite arbitrarily related meshes, we may then combine the efficiencies of adaptive refinements and of multigrid algorithms.