Two-level preconditioners for 2m'th order elliptic finite element problems

In this paper we analyze two-level preconditioners for second- and fourth-order elliptic boundary value problems. These preconditioners involve smoothing of the original problem and the solution (or preconditioning) of an auxiliary problem on a related mesh. Two abstract theorems are provided for this analysis. The properties needed to apply these theorems are developed for general finite element approximation spaces. These results are then applied to the second-order and biharmonic Dirichlet problems. Uniform preconditioning estimates are proved in the general case in which the triangulations are only assumed to be shape regular but not necessarily quasiuniform. When the meshes are of locally comparable size, this analysis applies to both conforming and nonconforming finite element approximations. When the preconditioning mesh is genuinely coarser than the original one, the analysis is given for the case in which the auxiliary problem is conforming. For this application we show that the appropriate smoothers can be defined by overlapping Schwarz methods.

Two-level preconditioners for 2m'th order elliptic finite element problems Academic Article