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Domain Decomposition Methods
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Domain Decomposition Methods

A special issue of Mathematical and Computational Applications (ISSN 2297-8747).

Deadline for manuscript submissions: closed (15 September 2021) | Viewed by 7012

Special Issue Editor

Prof. Dr. Victorita Dolean

E-Mail Website
Guest Editor
Department of Mathematics and Statistics, University of Strathclyde, 16 Richmond Street, Glasgow G1 1XQ, UK; LJAD, CNRS, University Côte d’Azur, 28 Avenue de Valrose, 06108 Nice CEDEX 2, France
Interests: applied mathematics; numerical analysis; scientific computing; high performance computing

Special Issue Information

Dear Colleagues,

Mathematical modelling in science and engineering problems relies heavily on partial differential equations. Accurate discretization of such PDEs is very often required, and this usually leads to potentially very large linear systems that must be solved in parallel. The computational resources (in terms of hardware) and computational time available can limit the high-fidelity of these simulations. With the advent of parallel computers and the availability of large computational clusters, algorithmic improvements are key in reducing the computational time and increasing the model complexity and accuracy. One of the success stories of parallel computing is linear solvers, but also hybrid solvers, like domain decomposition methods.

Contributions related to the development and analysis of the domain decomposition solvers with their different aspects (linear or non-linear, multilevel methods, scalability, HPC implementation, coupling of mathematical models, and computational challenges of large-scale problems) are welcome in this Special Issue.

Authors are also invited to submit any other relevant complementary materials, such as software or available links illustrating their research.

Prof. Dr. Victorita Dolean
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All papers will be peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematical and Computational Applications is an international peer-reviewed open access quarterly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Published Papers (3 papers)

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Research

22 pages, 1089 KiB  
Article
On the Convergence of the Damped Additive Schwarz Methods and the Subdomain Coloring
by Lori Badea
Math. Comput. Appl. 2022, 27(4), 59; https://doi.org/10.3390/mca27040059 - 13 Jul 2022
Viewed by 1193
Abstract
In this paper, we consider that the subdomains of the domain decomposition are colored such that the subdomains with the same color do not intersect and introduce and analyze the convergence of a damped additive Schwarz method related to such a subdomain coloring [...] Read more.
In this paper, we consider that the subdomains of the domain decomposition are colored such that the subdomains with the same color do not intersect and introduce and analyze the convergence of a damped additive Schwarz method related to such a subdomain coloring for the resolution of variational inequalities and equations. In this damped method, a single damping value is associated with all the subdomains having the same color. We first make this analysis both for variational inequalities and, as a special case, for equations in an abstract framework. By introducing an assumption on the decomposition of the convex set of the variational inequality, we theoretically analyze in a reflexive Banach space the convergence of the damped additive Schwarz method. The introduced assumption contains a constant C0, and we explicitly write the expression of the convergence rates, depending on the number of colors and the constant C0, and find the values of the damping constants which minimize them. For problems in the finite element spaces, we write the constant C0 as a function of the overlap parameter of the domain decomposition and the number of colors of the subdomains. We show that, for a fixed overlap parameter, the convergence rate, as a function of the number of subdomains has an upper limit which depends only on the number of the colors of the subdomains. Obviously, this limit is independent of the total number of subdomains. Numerical results are in agreement with the theoretical ones. They have been performed for an elasto-plastic problem to verify the theoretical predictions concerning the choice of the damping parameter, the dependence of the convergence on the overlap parameter and on the number of subdomains. Full article
(This article belongs to the Special Issue Domain Decomposition Methods)
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23 pages, 3275 KiB  
Article
Can DtN and GenEO Coarse Spaces Be Sufficiently Robust for Heterogeneous Helmholtz Problems?
by Niall Bootland and Victorita Dolean
Math. Comput. Appl. 2022, 27(3), 35; https://doi.org/10.3390/mca27030035 - 21 Apr 2022
Viewed by 2047
Abstract
Numerical solutions of heterogeneous Helmholtz problems present various computational challenges, with descriptive theory remaining out of reach for many popular approaches. Robustness and scalability are key for practical and reliable solvers in large-scale applications, especially for large wave number problems. In this work, [...] Read more.
Numerical solutions of heterogeneous Helmholtz problems present various computational challenges, with descriptive theory remaining out of reach for many popular approaches. Robustness and scalability are key for practical and reliable solvers in large-scale applications, especially for large wave number problems. In this work, we explore the use of a GenEO-type coarse space to build a two-level additive Schwarz method applicable to highly indefinite Helmholtz problems. Through a range of numerical tests on a 2D model problem, discretised by finite elements on pollution-free meshes, we observe robust convergence, iteration counts that do not increase with the wave number, and good scalability of our approach. We further provide results showing a favourable comparison with the DtN coarse space. Our numerical study shows promise that our solver methodology can be effective for challenging heterogeneous applications. Full article
(This article belongs to the Special Issue Domain Decomposition Methods)
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24 pages, 4238 KiB  
Article
Learning Adaptive Coarse Spaces of BDDC Algorithms for Stochastic Elliptic Problems with Oscillatory and High Contrast Coefficients
by Eric Chung, Hyea-Hyun Kim, Ming-Fai Lam and Lina Zhao
Math. Comput. Appl. 2021, 26(2), 44; https://doi.org/10.3390/mca26020044 - 06 Jun 2021
Cited by 1 | Viewed by 2561
Abstract
In this paper, we consider the balancing domain decomposition by constraints (BDDC) algorithm with adaptive coarse spaces for a class of stochastic elliptic problems. The key ingredient in the construction of the coarse space is the solutions of local spectral problems, which depend [...] Read more.
In this paper, we consider the balancing domain decomposition by constraints (BDDC) algorithm with adaptive coarse spaces for a class of stochastic elliptic problems. The key ingredient in the construction of the coarse space is the solutions of local spectral problems, which depend on the coefficient of the PDE. This poses a significant challenge for stochastic coefficients as it is computationally expensive to solve the local spectral problems for every realization of the coefficient. To tackle this computational burden, we propose a machine learning approach. Our method is based on the use of a deep neural network (DNN) to approximate the relation between the stochastic coefficients and the coarse spaces. For the input of the DNN, we apply the Karhunen–Loève expansion and use the first few dominant terms in the expansion. The output of the DNN is the resulting coarse space, which is then applied with the standard adaptive BDDC algorithm. We will present some numerical results with oscillatory and high contrast coefficients to show the efficiency and robustness of the proposed scheme. Full article
(This article belongs to the Special Issue Domain Decomposition Methods)
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