Bramble, J. H.; Pasciak, J. E.; Zhang, Xuejun Two-level preconditioners for \(2m\)’th order elliptic finite element problems. (English) Zbl 0860.65120 East-West J. Numer. Math. 4, No. 2, 99-120 (1996). Two-level techniques for constructing the preconditioners for the discrete systems which arise from the finite element approximation of second- and fourth-order elliptic boundary value problems are considered. A two-level preconditioner can be thought of as an additive version of the general two-level multigrid algorithm. Thus, the preconditioner for a problem on a given grid is constructed in terms of a solution or an approximate solution of an auxiliary problem on a related grid and a ‘smoother’ on the original grid.Two abstract theorems are provided. 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. Also, the authors provide a theorem which shows that the natural additive two-level preconditioner gives rise to uniformly convergent iterations under a regularity and approximation assumption. Reviewer: P.ChocholatĂ˝ (Bratislava) Cited in 2 Documents MSC: 65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations 65F35 Numerical computation of matrix norms, conditioning, scaling 35J40 Boundary value problems for higher-order elliptic equations 31A30 Biharmonic, polyharmonic functions and equations, Poisson’s equation in two dimensions 65F10 Iterative numerical methods for linear systems Keywords:uniform convergence; preconditioners; finite element; fourth-order elliptic boundary value problems; multigrid algorithm; biharmonic Dirichlet problems PDFBibTeX XMLCite \textit{J. H. Bramble} et al., East-West J. Numer. Math. 4, No. 2, 99--120 (1996; Zbl 0860.65120)