Adaptive selection of face coarse degrees of freedom in the BDDC and the FETI-DP iterative substructuring methods

Mandel, Jan; Sousedík, Bedřich

doi:10.1016/j.cma.2006.03.010

Computer Methods in Applied Mechanics and Engineering

2007

DOI: 10.1016/j.cma.2006.03.010

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Adaptive selection of face coarse degrees of freedom in the BDDC and the FETI-DP iterative substructuring methods

Jan Mandel

Bedřich Sousedík

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Cited by 107 publications

(160 citation statements)

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“…Then, for FETI-1 and BDD methods, a related adaptive approach was introduced [20]. At about the same time, published in [15], an adaptive approach for FETI-DP and BDDC methods was introduced by replacing a Poincaré inequality and an extension theorem (on edges) by eigenvalue problems; see [13] for the complete theory in 2D.Only recently, in 2015, for the first time a rigorous condition number estimate was then proven in [14] for the widely used adaptive approach from [17], for 2D problems. Also in [14], a comparison of three adaptive coarse spaces for 2D problems, i.e., those of [17], [4,12], and [13,15], was provided, discussing their strengths and weaknesses, and considering their performance in numerical experiments.…”

mentioning

confidence: 99%

“…Only recently, in 2015, for the first time a rigorous condition number estimate was then proven in [14] for the widely used adaptive approach from [17], for 2D problems. Also in [14], a comparison of three adaptive coarse spaces for 2D problems, i.e., those of [17], [4,12], and [13,15], was provided, discussing their strengths and weaknesses, and considering their performance in numerical experiments.…”

mentioning

confidence: 99%

“…In 2007, in [17], adaptive coarse spaces for FETI-DP and BDDC domain decomposition methods were proposed for 2D problems, at this time without a theoretical bound. Later, an adaptive coarse space for additive Schwarz methods was proposed [7,8], based on eigenvalue problems on complete subdomains, replacing a Poincaré estimate.…”

mentioning

confidence: 99%

“…Also in [14], a comparison of three adaptive coarse spaces for 2D problems, i.e., those of [17], [4,12], and [13,15], was provided, discussing their strengths and weaknesses, and considering their performance in numerical experiments. Then, but for an improved coarse space, in [11], a condition number estimate for 3D was shown for the first time; see also a remark in [5].…”

mentioning

confidence: 99%

“…Later, an adaptive coarse space for additive Schwarz methods was proposed [7,8], based on eigenvalue problems on complete subdomains, replacing a Poincaré estimate. In [18], the strategy from [17] was extended to 3D and used very successfully, also in a parallel implementation. Later, coarse spaces based on local Dirichlet-to-Neumann maps were introduced for Schwarz preconditioners [6].…”

mentioning

confidence: 99%

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Using Local Spectral Information in Domain Decomposition Methods – A Brief Overview in a Nutshell

Klawonn

Kühn

Rheinbach

2016

Proc Appl Math and Mech

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For second-order elliptic partial differential equations large discontinuities in the coefficients yield ill-conditioned stiffness matrices. The convergence of domain decomposition methods (DDM) can be improved by incorporating (numerically computed) local eigenvectors into the coarse space. Different adaptive coarse spaces for DDM have been constructed and used successfully. For two-level Schwarz, FETI-1 and BDD methods, adaptive coarse spaces with a rigorous theoretical basis are known for 2D and 3D. Although successfully in use for almost a decade, a theory for adaptive coarse spaces for FETI-DP and BDDC was lacking. While the problem was recently settled for 2D, the estimate for the 3D adaptive algorithm required improved coarse spaces. We give an brief overview of the literature, i.e., the different known approaches, and show numerical results for a specific adaptive FETI-DP method in 3D, where the condition number bound could only recently be proven. Domain decomposition methods (DDM) are fast parallel iterative solution methods for the solution of implicit linear or linearized systems from the discretization of partial differential equations (PDEs). The convergence theory of these methods typically relies on a (global) condition number bound constructed from local theoretical estimates for finite element functions. Recently, new approaches have gained interest where these estimates are replaced by computable spectral bounds, e.g., from discrete local eigenvalue problems. A corresponding enrichment of the coarse problem then yields a global bound which only depends on a user-defined tolerance and some geometric bounds, e.g., the maximum number of edges or faces of a subdomain. Thus, convergence is not affected by ill-conditioning, e.g., from heterogeneities or (in some approaches) from almost incompressibility.Already in [2], spectral information, i.e., the eigenvectors corresponding to the smallest eigenvalues of subdomain matrices, has been used with Neumann-Neumann methods, heuristically. In 2007, in [17], adaptive coarse spaces for FETI-DP and BDDC domain decomposition methods were proposed for 2D problems, at this time without a theoretical bound. Later, an adaptive coarse space for additive Schwarz methods was proposed [7,8], based on eigenvalue problems on complete subdomains, replacing a Poincaré estimate. In [18], the strategy from [17] was extended to 3D and used very successfully, also in a parallel implementation. Later, coarse spaces based on local Dirichlet-to-Neumann maps were introduced for Schwarz preconditioners [6]. Then, for FETI-1 and BDD methods, a related adaptive approach was introduced [20]. At about the same time, published in [15], an adaptive approach for FETI-DP and BDDC methods was introduced by replacing a Poincaré inequality and an extension theorem (on edges) by eigenvalue problems; see [13] for the complete theory in 2D.Only recently, in 2015, for the first time a rigorous condition number estimate was then proven in [14] for the widely used adaptive approach from [17], fo...

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confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Using Local Spectral Information in Domain Decomposition Methods – A Brief Overview in a Nutshell

Klawonn

Kühn

Rheinbach

2016

Proc Appl Math and Mech

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Combining machine learning and domain decomposition methods for the solution of partial differential equations—A review

et al. 2021

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Scientific machine learning (SciML), an area of research where techniques from machine learning and scientific computing are combined, has become of increasing importance and receives growing attention. Here, our focus is on a very specific area within SciML given by the combination of domain decomposition methods (DDMs) with machine learning techniques for the solution of partial differential equations. The aim of the present work is to make an attempt of providing a review of existing and also new approaches within this field as well as to present some known results in a unified framework; no claim of completeness is made. As a concrete example of machine learning enhanced DDMs, an approach is presented which uses neural networks to reduce the computational effort in adaptive DDMs while retaining their robustness. More precisely, deep neural networks are used to predict the geometric location of constraints which are needed to define a robust coarse space. Additionally, two recently published deep domain decomposition approaches are presented in a unified framework. Both approaches use physics‐constrained neural networks to replace the discretization and solution of the subdomain problems of a given decomposition of the computational domain. Finally, a brief overview is given of several further approaches which combine machine learning with ideas from DDMs to either increase the performance of already existing algorithms or to create completely new methods.

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BDDC for mixed‐hybrid formulation of flow in porous media with combined mesh dimensions

Šístek

Březina

Sousedík

2015

Numerical Linear Algebra App

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SUMMARYWe extend the Balancing Domain Decomposition by Constraints (BDDC) method to flows in porous media discretised by mixed-hybrid finite elements with combined mesh dimensions. Such discretisations appear when major geological fractures are modelled by 1D or 2D elements inside three-dimensional domains. In this set-up, the global problem as well as the substructure problems have a symmetric saddle-point structure, containing a 'penalty' block due to the combination of meshes. We show that the problem can be reduced by means of iterative substructuring to an interface problem, which is symmetric and positive definite. The interface problem can thus be solved by conjugate gradients with the BDDC method as a preconditioner. A parallel implementation of this algorithm is incorporated into an existing software package for subsurface flow simulations. We study the performance of the iterative solver on several academic and real-world problems. Numerical experiments illustrate its efficiency and scalability. Preprint version.

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Adaptive selection of face coarse degrees of freedom in the BDDC and the FETI-DP iterative substructuring methods

Cited by 107 publications

References 21 publications

Using Local Spectral Information in Domain Decomposition Methods – A Brief Overview in a Nutshell

Using Local Spectral Information in Domain Decomposition Methods – A Brief Overview in a Nutshell

Combining machine learning and domain decomposition methods for the solution of partial differential equations—A review

BDDC for mixed‐hybrid formulation of flow in porous media with combined mesh dimensions

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