English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Algebraic multigrid (AMG) is known to be an effective solver for many sparsesymmetric positive definite (SPD) linear systems. For SPD systems, theconvergence theory of AMG is well-understood in terms of the $A$-norm but in anonsymmetric setting such an energy norm is non-existent. For this reason,convergence of AMG for nonsymmetric systems of equations remains an open areaof research. Existing nonsymmetric AMG algorithms in this setting mostly relyon heuristics motivated by SPD convergence theory. In the SPD setting, theclassical form of optimal AMG interpolation provides a useful insight indetermining the two grid convergence rate of the method. In this work, wediscuss a generalization of the optimal AMG convergence theory targetingnonsymmetric problems by constructing a $2\times 2$ block symmetric indefinitesystem so that the Petrov-Galerkin AMG process for the nonsymmetric matrix $A$can be recast as a Galerkin AMG process for a symmetric indefinite system. Weshow that using this generalization of the optimal interpolation theory, onecan obtain the same identity for the two-grid convergence rate as that derivedin the SPD setting for optimal interpolation. We also provide supportingnumerical results for the convergence result and nonsymmetricadvection-diffusion problems.

Generalized Optimal AMG Convergence Theory for Nonsymmetric and Indefinite Problems