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Meshless Methods with Radial Basis Functions
Published in Jichun Li, Yi-Tung Chen, Computational Partial Differential Equations Using MATLAB®, 2019
The earliest concept of the domain decomposition method was introduced as a classical Schwarz alternating algorithm by Schwarz in 1870, which provided a fast and robust algorithm for solving the linear systems resulting from discretizations of PDEs. With the development of modern supercomputers, DDM has become a common tool for solving linear systems of equations. For a thorough theory on DDM and its applications, readers can consult some review papers [6, 41], books [57, 60, 61], and the proceedings of the international conferences on DDM (see www.ddm.org).
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.
Parallel Domain Decomposition of a FEM-based Tool for Numerical Modelling Mineral Slurry-like Flows
Published in International Journal of Computational Fluid Dynamics, 2022
Sergio Peralta, Jhon Córdova, Cesar Celis, Danmer Maza
Domain decomposition methods (DDM) consist in decomposing a computational domain and distributing the associated work in parallel environments (Houzeaux et al. 2017). Currently, DDM are the most common approaches used in the development of FEM based parallel flow solvers (Reddy and Gartling 2010). A comprehensive review of DDM related applications, including multiphase or non-Newtonian flows, is presented in Tang, Haynes, and Houzeaux (2021). About 400 references describing DDM efforts carried out in the past 40 years were reviewed in the referred work. Notice as well that in literature direct or iterative solvers coupled with DDM are reported. Direct solvers are usually very robust and consume large amounts of memory. In contrast, iterative solvers are highly problem dependent and memory efficient (Raju and Khaitann.d.). For instance, parallelism features of a finite element-based code using DDM and a sparse direct solver are discussed by Raju and Khaitan (n.d.). In that work, a Newton's iterative loop coupled with an additive Schwarz one was used to solve the Navier–Stokes equations. The developed algorithm was found to scale well in terms of computational time and memory requirements. In turn, a stabilised finite element method for the 3D non-Newtonian Navier–Stokes equations and a parallel domain decomposition one for solving the sparse system was introduced in Shiu, Hwang, and Cai (2015). The highly nonlinear equations system resulting from non-Newtonian effects was solved there by a purely iterative Newton–Krylov–Schwarz algorithm. Superlinear speedup was obtained using up to 512 processors (Shiu, Hwang, and Cai 2015). In the literature, traditional parallelisation methods such as those based of the parallel sparse matrix-vector product (SpMV) are also discussed. Notice that parallel SpMV formulations are built in such a way that the solutions obtained are the same as those produced by their sequential counterparts, without resorting to iterative processes or sacrificing accuracy. For instance, Houzeaux et al. (2018) extended a parallel SpMV to glue the solution of non-matching (non-conforming) meshes through the introduction of transmission matrices. This SpMV extension enabled the implicit and parallel solution of partial differential equations on non-matching meshes and the implicit coupling of multiphysics problems. Finally, notice that DDM related algorithms can be based on iterative updates of the boundary conditions imposed on the interfaces between subdomains, the so-called transmission conditions (Houzeaux and Codina 2001). For further details about these transmission conditions, the interested reader may refer to past works dealing with them in complex flows applications (Houzeaux and Codina 2003; Japhet et al. 2017; Martin 2009). Notice that, in some of the references highlighted above, the transmission conditions were not necessarily intended for parallelisation purposes. However, these works have been reviewed here because they are relevant to the development of transmission conditions such as those discussed here.