Algebraic multigrid theory: The symmetric case
Applied Mathematics and Computation - Second Copper Mountain conference on Multigrid methods Copper Mountain, Colorado
Multigrid convergence for nonsymmetric, indefinite variational problems on one smoothing step
Applied Mathematics and Computation - Second Copper Mountain conference on Multigrid methods Copper Mountain, Colorado
An algebraic theory for multigrid methods for variational problems
SIAM Journal on Numerical Analysis
Convergence analysis of multigrid algorithms for nonselfadjoint and indefinite elliptic problems
SIAM Journal on Numerical Analysis
Uniform convergence of multigrid V-cycle iterations for indefinite and nonsymmetric problems
SIAM Journal on Numerical Analysis - Special issue: the articles in this issue are dedicated to Seymour V. Parter
Fast Multigrid Solution of the Advection Problem with Closed Characteristics
SIAM Journal on Scientific Computing
Coarse-Grid Correction for Nonelliptic and Singular Perturbation Problems
SIAM Journal on Scientific Computing
A multigrid tutorial (2nd ed.)
A multigrid tutorial (2nd ed.)
Algebraic Multigrid on Unstructured Meshes
Algebraic Multigrid on Unstructured Meshes
An Aggregation Multigrid Solver for convection-diffusion problems onunstructured meshes.
An Aggregation Multigrid Solver for convection-diffusion problems onunstructured meshes.
Adaptive Smoothed Aggregation ($\alpha$SA)
SIAM Journal on Scientific Computing
A New Petrov-Galerkin Smoothed Aggregation Preconditioner for Nonsymmetric Linear Systems
SIAM Journal on Scientific Computing
Smoothed Aggregation Multigrid for Markov Chains
SIAM Journal on Scientific Computing
Least-Squares Finite Element Methods for Quantum Electrodynamics
SIAM Journal on Scientific Computing
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Applying smoothed aggregation (SA) multigrid to solve a nonsymmetric linear system, $A\mathbf{x} =\mathbf{b}$, is often impeded by the lack of a minimization principle that can be used as a basis for the coarse-grid correction process. This paper proposes a Petrov-Galerkin (PG) approach based on applying SA to either of two symmetric positive definite (SPD) matrices, $\sqrt{A^{t}A}$ or $\sqrt{AA^{t}}$. These matrices, however, are typically full and difficult to compute, so it is not computationally efficient to use them directly to form a coarse-grid correction. The proposed approach approximates these coarse-grid corrections by using SA to accurately approximate the right and left singular vectors of $A$ that correspond to the lowest singular value. These left and right singular vectors are used to construct the restriction and interpolation operators, respectively. A preliminary two-level convergence theory is presented, suggesting that more relaxation should be applied than for an SPD problem. Additionally, a nonsymmetric version of adaptive SA ($\alpha$SA) is given that automatically constructs SA multigrid hierarchies using a stationary relaxation process on all levels. Numerical results are reported for convection-diffusion problems in two dimensions with varying amounts of convection for constant, variable, and recirculating convection fields. The results suggest that the proposed approach is algorithmically scalable for problems coming from these nonsymmetric scalar PDEs (with the exception of recirculating flow). This paper serves as a first step for nonsymmetric $\alpha$SA. The long-term goal of this effort is to develop nonsymmetric $\alpha$SA for systems of PDEs, where the SA framework has proven to be well suited for adaptivity in SPD problems.