Algebraic multigrid for stabilized finite element discretizations of the Navier Stokes equation

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2002.Includes bibliographical references (p. 141-152).A multilevel method for the solution of systems of equations generated by stabilized Finite Element discretizations of the Euler and Navier Stokes equations on generalized unstructured grids is described. The method is based on an elemental agglomeration multigrid which produces a hierarchical sequence of coarse subspaces. Linear combinations of the basis functions from a given space form the next subspace and the use of the Galerkin Coarse Grid Approximation (GCA) within an Algebraic Multigrid (AMG) context properly defines the hierarchical sequence. The multigrid coarse spaces constructed by the elemental agglomeration algorithm are based on a semi-coarsening scheme designed to reduce grid anisotropy. The multigrid transfer operators are induced by the graph of the coarse space mesh and proper consideration is given to the boundary conditions for an accurate representation of the coarse space operators. A generalized line implicit relaxation scheme is also described where the lines are constructed to follow the direction of strongest coupling. The solution algorithm is motivated by the decomposition of the system characteristics into acoustic and convective modes. Analysis of the application of elemental agglomeration AMG (AMGe) to stabilized numerical schemes shows that a characteristic length based rescaling of the numerical stabilization is necessary for a consistent multigrid representation.by Tolulope Olawale Okusanya.Ph.D

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..