1,927 research outputs found
Nonlinear multigrid based on local spectral coarsening for heterogeneous diffusion problems
This work develops a nonlinear multigrid method for diffusion problems
discretized by cell-centered finite volume methods on general unstructured
grids. The multigrid hierarchy is constructed algebraically using aggregation
of degrees of freedom and spectral decomposition of reference linear operators
associated with the aggregates. For rapid convergence, it is important that the
resulting coarse spaces have good approximation properties. In our approach,
the approximation quality can be directly improved by including more spectral
degrees of freedom in the coarsening process. Further, by exploiting local
coarsening and a piecewise-constant approximation when evaluating the nonlinear
component, the coarse level problems are assembled and solved without ever
re-visiting the fine level, an essential element for multigrid algorithms to
achieve optimal scalability. Numerical examples comparing relative performance
of the proposed nonlinear multigrid solvers with standard single-level
approaches -- Picard's and Newton's methods -- are presented. Results show that
the proposed solver consistently outperforms the single-level methods, both in
efficiency and robustness
Some Comments on Multigrid Methods for Computing Propagators
I make three conceptual points regarding multigrid methods for computing
propagators in lattice gauge theory: 1) The class of operators handled by the
algorithm must be stable under coarsening. 2) Problems related by symmetry
should have solution methods related by symmetry. 3) It is crucial to
distinguish the vector space from its dual space . All the existing
algorithms violate one or more of these principles.Comment: 16 pages, LaTeX plus subeqnarray.sty (included at end),
NYU-TH-93/07/0
HP-multigrid as smoother algorithm for higher order discontinuous Galerkin discretizations of advection dominated flows. Part II. Optimization of the Runge-Kutta smoother
Using a detailed multilevel analysis of the complete hp-Multigrid as Smoother algorithm accurate predictions are obtained of the spectral radius and operator norms of the multigrid error transformation operator. This multilevel analysis is used to optimize the coefficients in the semi-implicit Runge-Kutta smoother, such that the spectral radius of the multigrid error transformation operator is minimal under properly chosen constraints. The Runge-Kutta coefficients for a wide range of cell Reynolds numbers and a detailed analysis of the performance of the hp-MGS algorithm are presented. In addition, the computational complexity of the hp-MGS algorithm is investigated. The hp-MGS algorithm is tested on a fourth order accurate space-time discontinuous Galerkin finite element discretization of the advection-diffusion equation for a number of model problems, which include thin boundary layers and highly stretched meshes, and a non-constant advection velocity. For all test cases excellent multigrid convergence is obtained
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