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Numerical Methods for PDEs
Published in Daniel Zwillinger, Vladimir Dobrushkin, Handbook of Differential Equations, 2021
Daniel Zwillinger, Vladimir Dobrushkin
The procedure illustrated in this section is called Schwarz's method, it is only one of several different domain decomposition methods. See the book by Dolean et al.[363], which is available on-line.
Modeling/simulation of transient linear heat conduction problems via integrating a wide variety of space/time methods and choices
Published in Numerical Heat Transfer, Part B: Fundamentals, 2023
Domain decomposition methods have been widely used in numerical analysis in which the single domain approach is not able to describe the physics of the entire system [6–8]. For complex multi-physical problems, domain decomposition methods provide a viable way to simplify the problem and reduce the complexity. In general, two adjacent regions may be governed by different partial differential equations, such as multi-body dynamics coupling two solid regions and fluid-solid interaction coupling solid and fluid, which has to be split into two parts [9, 10]. Also, there exist situations wherein the same governing equation is used to describe physics of the entire system while the material properties are varied greatly such that an ill-conditioned stiffness matrix arises in the numerical simulation. To address this issue, the original problem domain can be decomposed into several smaller subdomains that can be isolated and evaluated independently, and then coupled together [11, 12].
Newton-type multilevel optimization method
Published in Optimization Methods and Software, 2022
Chin Pang Ho, Michal Kočvara, Panos Parpas
We point out that this idea of using multiple coarse models is related to domain decomposition methods, which solve (non-)linear equations arising from PDEs. Domain decomposition methods partition the problem domain into several subdomains, and thus decompose the original problem into several smaller problems. We refer the reader to [7] for more details about domain decomposition methods.