Publications

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In recent years, considerable effort has been placed on developing efficient and robust solution algorithms for the incompressible Navier-Stokes equations based on preconditioned Krylov methods. These include physics-based methods, such as SIMPLE, and purely algebraic preconditioners based on the approximation of the Schur complement. All these techniques can be represented as approximate block factorization (ABF) type preconditioners. The goal is to decompose the application of the preconditioner into simplified sub-systems in which scalable multi-level type solvers can be applied. In this paper we develop a taxonomy of these ideas based on an adaptation of a generalized approximate factorization of the Navier-Stokes system first presented in [25]. This taxonomy illuminates the similarities and differences among these preconditioners and the central role played by efficient approximation of certain Schur complement operators. We then present a parallel computational study that examines the performance of these methods and compares them to an additive Schwarz domain decomposition (DD) algorithm. Results are presented for two and three-dimensional steady state problems for enclosed domains and inflow/outflow systems on both structured and unstructured meshes. The numerical experiments are performed using MPSalsa, a stabilized finite element code.

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We report on algebraic multilevel preconditioners for the parallel solution of linear systems arising from a Newton procedure applied to the finite-element (FE) discretization of the incompressible Navier-Stokes equations. We focus on the issue of how to coarsen FE operators produced from high aspect ratio elements.

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The goal of our paper is to compare a number of algorithms for computing a large number of eigenvectors of the generalized symmetric eigenvalue problem arising from a modal analysis of elastic structures. The shift-invert Lanczos algorithm has emerged as the workhorse for the solution of this generalized eigenvalue problem; however, a sparse direct factorization is required for the resulting set of linear equations. Instead, our paper considers the use of preconditioned iterative methods. We present a brief review of available preconditioned eigensolvers followed by a numerical comparison on three problems using a scalable algebraic multigrid (AMG) preconditioner. Copyright © 2005 John Wiley & Sons, Ltd.

In this paper we present a two-level overlapping domain decomposition preconditioner for the finite-element discretization of elliptic problems in two and three dimensions. The computational domain is partitioned into overlapping subdomains, and a coarse space correction, based on aggregation techniques, is added. Our definition of the coarse space does not require the introduction of a coarse grid. We consider a set of assumptions on the coarse basis functions to bound the condition number of the resulting preconditioned system. These assumptions involve only geometrical quantities associated with the aggregates and the subdomains. We prove that the condition number using the two-level additive Schwarz preconditioner is O(H/{delta} + H{sub 0}/{delta}), where H and H{sub 0} are the diameters of the subdomains and the aggregates, respectively, and {delta} is the overlap among the subdomains and the aggregates. This extends the bounds presented in [C. Lasser and A. Toselli, Convergence of some two-level overlapping domain decomposition preconditioners with smoothed aggregation coarse spaces, in Recent Developments in Domain Decomposition Methods, Lecture Notes in Comput. Sci. Engrg. 23, L. Pavarino and A. Toselli, eds., Springer-Verlag, Berlin, 2002, pp. 95-117; M. Sala, Domain Decomposition Preconditioners: Theoretical Properties, Application to the Compressible Euler Equations, Parallel Aspects, Ph.D. thesis, Ecole Polytechnique Federale de Lausanne, Lausanne, Switzerland, 2003; M. Sala, Math. Model. Numer. Anal., 38 (2004), pp. 765-780]. Numerical experiments on a model problem are reported to illustrate the performance of the proposed preconditioner.

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The solution of the governing steady transport equations for momentum, heat and mass transfer in fluids undergoing non-equilibrium chemical reactions can be extremely challenging. The difficulties arise from both the complexity of the nonlinear solution behavior as well as the nonlinear, coupled, non-symmetric nature of the system of algebraic equations that results from spatial discretization of the PDEs. In this paper, we briefly review progress on developing a stabilized finite element (FE) capability for numerical solution of these challenging problems. The discussion considers the stabilized FE formulation for the low Mach number Navier-Stokes equations with heat and mass transport with non-equilibrium chemical reactions, and the solution methods necessary for detailed analysis of these complex systems. The solution algorithms include robust nonlinear and linear solution schemes, parameter continuation methods, and linear stability analysis techniques. Our discussion considers computational efficiency, scalability, and some implementation issues of the solution methods. Computational results are presented for a CFD benchmark problem as well as for a number of large-scale, 2D and 3D, engineering transport/reaction applications.

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The solution of the governing steady transport equations for momentum, heat and mass transfer in fluids undergoing non-equilibrium chemical reactions can be extremely challenging. The difficulties arise from both the complexity of the nonlinear solution behavior as well as the nonlinear, coupled, non-symmetric nature of the system of algebraic equations that results from spatial discretization of the PDEs. In this paper, we briefly review progress on developing a stabilized finite element ( FE) capability for numerical solution of these challenging problems. The discussion considers the stabilized FE formulation for the low Mach number Navier-Stokes equations with heat and mass transport with non-equilibrium chemical reactions, and the solution methods necessary for detailed analysis of these complex systems. The solution algorithms include robust nonlinear and linear solution schemes, parameter continuation methods, and linear stability analysis techniques. Our discussion considers computational efficiency, scalability, and some implementation issues of the solution methods. Computational results are presented for a CFD benchmark problem as well as for a number of large-scale, 2D and 3D, engineering transport/reaction applications.

This study investigates algebraic multilevel domain decomposition preconditioners of the Schwarz type for solving linear systems associated with Newton-Krylov methods. The key component of the preconditioner is a coarse approximation based on algebraic multigrid ideas to approximate the global behavior of the linear system. The algebraic multilevel preconditioner is based on an aggressive coarsening graph partitioning of the non-zero block structure of the Jacobian matrix. The scalability of the preconditioner is presented as well as comparisons with a two-level Schwarz preconditioner using a geometric coarse grid operator. These comparisons are obtained on large-scale distributed-memory parallel machines for systems arising from incompressible flow and transport using a stabilized finite element formulation. The results demonstrate the influence of the smoothers and coarse level solvers for a set of 3D example problems. For preconditioners with more than one level, careful attention needs to be given to the balance of robustness and convergence rate for the smoothers and the cost of applying these methods. For properly chosen parameters, the two- and three-level preconditioners are demonstrated to be scalable to 1024 processors.

ML is a multigrid preconditioning package intended to solve linear systems of equations Ax = b where A is a user supplied n x n sparse matrix, b is a user supplied vector of length n and x is a vector of length n to be computed. ML should be used on large sparse linear systems arising from partial differential equation (PDE) discretizations. While technically any linear system can be considered, ML should be used on linear systems that correspond to things that work well with multigrid methods (e.g. elliptic PDEs). ML can be used as a stand-alone package or to generate preconditioners for a traditional iterative solver package (e.g. Krylov methods). We have supplied support for working with the Aztec 2.1 and AztecOO iterative package [16]. However, other solvers can be used by supplying a few functions. This document describes one specific algebraic multigrid approach: smoothed aggregation. This approach is used within several specialized multigrid methods: one for the eddy current formulation for Maxwell's equations, and a multilevel and domain decomposition method for symmetric and nonsymmetric systems of equations (like elliptic equations, or compressible and incompressible fluid dynamics problems). Other methods exist within ML but are not described in this document. Examples are given illustrating the problem definition and exercising multigrid options.