scholarly journals Convergence Rate of Overlapping Domain Decomposition Methods for the Rudin--Osher--Fatemi Model Based on a Dual Formulation

2015 ◽  
Vol 8 (1) ◽  
pp. 564-591 ◽  
Author(s):  
Huibin Chang ◽  
Xue-Cheng Tai ◽  
Li-Lian Wang ◽  
Danping Yang
Keyword(s):  
Convergence Rate ◽  
Domain Decomposition ◽  
Decomposition Methods ◽  
Domain Decomposition Methods ◽  
Dual Formulation ◽  
Model Based ◽  
Overlapping Domain Decomposition

scholarly journals Analysis of Patch Substructuring Methods

2007 ◽  
Vol 17 (3) ◽  
pp. 395-402 ◽  
Author(s):  
Martin Gander ◽  
Laurence Halpern ◽  
Frédéric Magoulès ◽  
François-Xavier Roux
Keyword(s):  
Convergence Rate ◽  
Numerical Experiments ◽  
Decomposition Methods ◽  
Transmission Conditions ◽  
New Method ◽  
Domain Decomposition Methods ◽  
Convergence Properties ◽  
Schwarz Method ◽  
Overlapping Domain Decomposition ◽  
Special Case

Analysis of Patch Substructuring MethodsPatch substructuring methods are non-overlapping domain decomposition methods like classical substructuring methods, but they use information from geometric patches reaching into neighboring subdomains, condensated on the interfaces, to enhance the performance of the method, while keeping it non-overlapping. These methods are very convenient to use in practice, but their convergence properties have not been studied yet. We analyze geometric patch substructuring methods for the special case of one patch per interface. We show that this method is equivalent to an overlapping Schwarz method using Neumann transmission conditions. This equivalence is obtained by first studying a new, algebraic patch method, which is equivalent to the classical Schwarz method with Dirichlet transmission conditions and an overlap corresponding to the size of the patches. Our results motivate a new method, the Robin patch method, which is a linear combination of the algebraic and the geometric one, and can be interpreted as an optimized Schwarz method with Robin transmission conditions. This new method has a significantly faster convergence rate than both the algebraic and the geometric one. We complement our results by numerical experiments.

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