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Preconditioned discontinuous Galerkin method and convection‐diffusion‐reaction problems with guaranteed bounds to resulting spectra.

Authors :
Gaynutdinova, Liya
Ladecký, Martin
Pultarová, Ivana
Vlasák, Miloslav
Zeman, Jan
Source :
Numerical Linear Algebra with Applications. Aug2024, Vol. 31 Issue 4, p1-17. 17p.
Publication Year :
2024

Abstract

This paper focuses on the design, analysis and implementation of a new preconditioning concept for linear second order partial differential equations, including the convection‐diffusion‐reaction problems discretized by Galerkin or discontinuous Galerkin methods. We expand on the approach introduced by Gergelits et al. and adapt it to the more general settings, assuming that both the original and preconditioning matrices are composed of sparse matrices of very low ranks, representing local contributions to the global matrices. When applied to a symmetric problem, the method provides bounds to all individual eigenvalues of the preconditioned matrix. We show that this preconditioning strategy works not only for Galerkin discretization, but also for the discontinuous Galerkin discretization, where local contributions are associated with individual edges of the triangulation. In the case of nonsymmetric problems, the method yields guaranteed bounds to real and imaginary parts of the resulting eigenvalues. We include some numerical experiments illustrating the method and its implementation, showcasing its effectiveness for the two variants of discretized (convection‐)diffusion‐reaction problems. [ABSTRACT FROM AUTHOR]

Details

Language :
English
ISSN :
10705325
Volume :
31
Issue :
4
Database :
Academic Search Index
Journal :
Numerical Linear Algebra with Applications
Publication Type :
Academic Journal
Accession number :
178441654
Full Text :
https://doi.org/10.1002/nla.2549