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An optimized Schwarz method with two‐sided Robin transmission conditions for the Helmholtz equation
Author(s) -
Gander M. J.,
Halpern L.,
Magoulès F.
Publication year - 2007
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.1433
Subject(s) - schwarz alternating method , additive schwarz method , convergence (economics) , mathematics , helmholtz free energy , helmholtz equation , derivative (finance) , transmission (telecommunications) , domain decomposition methods , mathematical optimization , mathematical analysis , computer science , finite element method , telecommunications , boundary value problem , physics , quantum mechanics , financial economics , economics , thermodynamics , economic growth
Optimized Schwarz methods are working like classical Schwarz methods, but they are exchanging physically more valuable information between subdomains and hence have better convergence behaviour. The new transmission conditions include also derivative information, not just function values, and optimized Schwarz methods can be used without overlap. In this paper, we present a new optimized Schwarz method without overlap in the 2d case, which uses a different Robin condition for neighbouring subdomains at their common interface, and which we call two‐sided Robin condition. We optimize the parameters in the Robin conditions and show that for a fixed frequency an asymptotic convergence factor of 1 – O ( h 1/4 ) in the mesh parameter h can be achieved. If the frequency is related to the mesh parameter h, h = O (1/ω γ ) for γ⩾1, then the optimized asymptotic convergence factor is 1 – O (ω (1–2γ)/8 ). We illustrate our analysis with 2d numerical experiments. Copyright © 2007 John Wiley & Sons, Ltd.