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An analysis of overlapping Schwarz method for a weakly coupled system of singularly perturbed convection‐diffusion equations
Author(s) -
Christy Roja J.,
Tamilselvan A.,
Geetha N.
Publication year - 2020
Publication title -
international journal for numerical methods in fluids
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.938
H-Index - 112
eISSN - 1097-0363
pISSN - 0271-2091
DOI - 10.1002/fld.4794
Subject(s) - mathematics , schwarz alternating method , convection–diffusion equation , convergence (economics) , norm (philosophy) , mathematical analysis , boundary (topology) , additive schwarz method , scheme (mathematics) , finite difference method , domain decomposition methods , finite element method , physics , political science , law , economics , thermodynamics , economic growth
Summary In this article, we have developed an overlapping Schwarz method for a weakly coupled system of convection‐diffusion equations. The method splits the original domain into two overlapping subdomains. A hybrid difference scheme is proposed in which on the boundary layer region, we use the central finite difference scheme on a uniform mesh, whereas on the nonlayer region, we use the mid‐point difference scheme on a uniform mesh. It is shown that the numerical approximations converge in the maximum norm to the exact solution. We have proved that, when appropriate subdomains are used, the method produces almost second‐order convergence. Furthermore, it is shown that two iterations are sufficient to achieve the expected accuracy. Numerical examples are presented to support the theoretical results. The main advantage of this method used with the proposed scheme is that it reduces iteration counts very much and easily identifies in which iteration the Schwarz iterate terminates.