Premium
Convergence bounds of GMRES with Schwarz' preconditioner for the scattering problem
Author(s) -
Ben Belgacem Faker,
Gmati Nabil,
Jelassi Faten
Publication year - 2009
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.2627
Subject(s) - preconditioner , generalized minimal residual method , mathematics , convergence (economics) , schwarz alternating method , mathematical proof , rate of convergence , symbolic convergence theory , polynomial , kernel (algebra) , mathematical optimization , mathematical analysis , iterative method , computer science , discrete mathematics , domain decomposition methods , finite element method , geometry , construct (python library) , physics , computer network , channel (broadcasting) , economics , thermodynamics , economic growth , programming language
We consider the Jacobi preconditioner of the GMRES method introduced by Liu and Jin for the scattering problem ( IEEE Trans. Ante. Prop. 2002; 50 :132–140). We explain why it is a particular form of the Schwarz' preconditioner with a complete overlap and specific transmission conditions. So far, a superlinear convergence has been predicted by the general theory without any additional indication on the convergence rates. Here, we establish error bounds that provide accurate convergence rates in two and three dimensions. Courant–Weyl's min–max principle applied to some kernel operators together with some polynomial approximation estimates are the milestones for the proofs. Copyright © 2009 John Wiley & Sons, Ltd.