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This paper develops ellipticity estimates and discretization error bounds for elliptic equations (with lower order terms) that are reformulated as a least{squares problem for an equivalent rst{order system. The main result is the proof of ellipticity, which is used in a companion paper to establish optimal convergence of multiplicative and additive solvers… (More)

- Z. CAI, S. F. MCCORMICK
- 1997

Following our earlier work on general second-order scalar equations, here we develop a least-squares functional for the two-and three-dimensional Stokes equations, generalized slightly by allowing a pressure term in the continuity equation. By introducing a velocity flux variable and associated curl and trace equations, we are able to establish ellipticity… (More)

We introduce AMGe, an algebraic multigrid method for solving the discrete equations that arise in Ritz-type nite element methods for partial diierential equations. Assuming access to the element stiiness matrices, AMGe is based on the use of two local measures, which are derived from global measures that appear in existing multigrid theory. These new… (More)

Algebraic multigrid (AMG) is currently undergoing a resurgence in popularity, due in part to the dramatic increase in the need to solve physical problems posed on very large, unstructured grids. While AMG has proved its usefulness on various problem types, it is not commonly understood how wide a range of applicability the method has. In this study, we… (More)

This paper develops a least-squares approach to the solution of the incompressible Navier–Stokes equations in primitive variables. As with our earlier work on Stokes equations, we recast the Navier–Stokes equations as a first-order system by introducing a velocity-flux variable and associated curl and trace equations. We show that a least-squares principle… (More)

- Multigrid, M. Brezina, R. Falgout, S. MacLachlan, T. Manteuffel, S. McCormick +1 other
- 2005

Substantial effort has been focused over the last two decades on developing multilevel iterative methods capable of solving the large linear systems encountered in engineering practice. These systems often arise from discretizing partial differential equations over unstructured meshes, and the particular parameters or geometry of the physical problem being… (More)

Spectral AMGe (ρAMGe), is a new algebraic multigrid method for solving discretizations that arise in Ritz-type finite element methods for partial differential equations. The method assumes access to the element stiffness matrices in order to lessen certain presumptions that can limit other algebraic methods. ρAMGe uses the spectral decomposition of small… (More)

This paper continues the development of the least-squares methodology for the solution of the incompressible Navier-Stokes equations started in Part I. Here we again use a velocity-flux first-order Navier-Stokes system, but our focus now is on a practical algorithm based on a discrete negative norm.

Bootstrap Algebraic Multigrid (BAMG) is a multigrid-based solver for matrix equations of the form Ax = b. Its aim is to automatically determine the interpolation weights used in algebraic multigrid (AMG) by locally fitting a set of test vectors that have been relaxed as solutions to the corresponding homogeneous equation, Ax = 0, and are then possibly… (More)