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On the convergence of basic iterative methods for convection–diffusion equations
Author(s) -
Bey Jürgen,
Reusken Arnold
Publication year - 1999
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/(sici)1099-1506(199907/08)6:5<329::aid-nla167>3.0.co;2-a
Subject(s) - mathematics , discretization , gauss–seidel method , diagonally dominant matrix , jacobi method , finite element method , convergence (economics) , iterative method , finite volume method , diagonal , successive over relaxation , stiffness matrix , mathematical analysis , mathematical optimization , local convergence , geometry , pure mathematics , invertible matrix , physics , mechanics , economics , thermodynamics , economic growth
In this paper we analyze convergence of basic iterative Jacobi and Gauss–Seidel type methods for solving linear systems which result from finite element or finite volume discretization of convection–diffusion equations on unstructured meshes. In general the resulting stiffness matrices are neither M‐matrices nor satisfy a diagonal dominance criterion. We introduce two newmatrix classes and analyse the convergence of the Jacobi and Gauss–Seidel methods for matrices from these classes. A new convergence result for the Jacobi method is proved and negative results for the Gauss–Seidel method are obtained. For a few well‐known discretization methods it is shown that the resulting stiffness matrices fall into the new matrix classes. Copyright © 1999 John Wiley & Sons, Ltd.