Multilevel Algebraic Elliptic Solvers

: We survey some of the recent research in developing multilevel algebraic solvers for elliptic problems. A key concept is the design of a hierarchy of coarse spaces and related interpolation operators which together satisfy certain approximation and stability properties to ensure the rapid convergence of the resulting multigrid algorithms. We will discuss smoothed agglomeration methods, harmonic extension methods, and global energy minimization methods for the construction of these coarse spaces and interpolation operators. 1 Introduction There has been a recent resurgence of interest in algebraic multilevel elliptic solvers. Part of the motivation for considering the multilevel approach is that it is the only general method for deriving scalable solvers for large problems. Another reason is that multilevel methods are often naturally parallelizable. The main motivation for considering algebraic solvers is ease of use as often the user need only supply the problem in purely matrix-ve..