BDDC and FETI-DP under minimalist assumptions

Mandel, Jan; Sousedík, Bedřich

doi:10.1007/s00607-007-0254-y

Cited by 24 publications

(35 citation statements)

References 22 publications

(58 reference statements)

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“…We note that (37) is related to (26)- (28) in the same way as (14) is related to (11)- (13). The main difference in the structure of the matrices between (37) and (14) is the additional block…”

Section: Modelling Of Fracturesmentioning

confidence: 96%

“…Equation (13) imposes a continuity condition on the normal component of the velocity (also called normal flux) u · n across F which guarantees that u ∈ V. This condition also implies the equivalence of the two formulations (7)-(8) and (11)- (13). We note that the Lagrange multipliers λ can be interpreted as the approximation of the trace of p on F, see [34] for details.…”

Section: Model Problemmentioning

confidence: 99%

“…It has been proposed independently in [10,11,12]; see [13,14] for the proof of equivalence. Even though BDDC has been originally formulated for elliptic problems, it has been successfully extended, for example, beyond elliptic cases [15,16] and to multiple levels [17,18].…”

Section: Introductionmentioning

confidence: 99%

“…In order to apply the existing BDDC theory for elliptic problems (e.g. [13,36]) to the proposed preconditioner, we introduce some additional notation and make a few observations. Due to (44), the substructure corrections in (60) can be written equivalently as…”

Section: Algorithmmentioning

confidence: 99%

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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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Section: Modelling Of Fracturesmentioning

confidence: 96%

Section: Model Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Algorithmmentioning

confidence: 99%

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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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“…This Balancing Domain Decomposition by Constraints (BDDC) method was later proven by Mandel et al [18] to have a condition number bound of κ ≤ C(1 + log(H/h)) 2 for preconditioned system of a continuous finite element discretization of second order elliptic problems. Further analysis of BDDC methods as well as the relationship between BDDC methods and dual-primal Finite Element Tearing and Interconnecting (FETI-DP) methods has been presented in [16,7,19].…”

mentioning

confidence: 99%

A Unified Analysis of Balancing Domain Decomposition by Constraints for Discontinuous Galerkin Discretizations

Diosady¹,

Darmofal²

2012

SIAM J. Numer. Anal.

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Abstract. The BDDC algorithm is extended to a large class of discontinuous Galerkin (DG) discretizations of second order elliptic problems. An estimate of C(1 + log(H/h)) 2 is obtained for the condition number of the preconditioned system where C is a constant independent of h or H or large jumps in the coefficient of the problem. Numerical simulations are presented which confirm the theoretical results. A key component for the development and analysis of the BDDC algorithm is a novel perspective presenting the DG discretization as the sum of element-wise "local" bilinear forms. The element-wise perspective allows for a simple unified analysis of a variety of DG methods and leads naturally to the appropriate choice for the subdomain-wise local bilinear forms. Additionally, this new perspective enables a connection to be drawn between the DG discretization and a related continuous finite element discretization to simplify the analysis of the BDDC algorithm.Key words. Discontinuous Galerkin, Domain Decomposition, BDDC AMS subject classifications. 65M55, 65M60, 65N30, 65N551. Introduction. Domain decomposition (DD) methods provide efficient parallel preconditioners for solving large system of equations arising from the discretization of partial differential equations. The development of domain decompositions methods for the solution of elliptic problems using conforming finite element methods has matured significantly over the past 20 years. Toselli and Widlund provide a detailed overview of domain decomposition methods in [22]. In this paper we consider a class of non-overlapping domain decomposition methods based on the Neummann-Neumann methods originally introduced by Bourgat et al. [6]. These methods were improved by introducing a coarse space based on the null-space of the local Schur complement problems, leading to the Balancing Domain Decomposition (BDD) method of Mandel [17]. Dohrmann extended the BDD method by selecting a coarse space formed by enforcing continuity of a small set of primal degrees of freedom [11]. This Balancing Domain Decomposition by Constraints (BDDC) method was later proven by Mandel et al [18] to have a condition number bound of κ ≤ C(1 + log(H/h)) 2 for preconditioned system of a continuous finite element discretization of second order elliptic problems. Further analysis of BDDC methods as well as the relationship between BDDC methods and dual-primal Finite Element Tearing and Interconnecting (FETI-DP) methods has been presented in [16,7,19].

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A local hybrid surrogate‐based finite element tearing interconnecting dual‐primal method for nonsmooth random partial differential equations

Eigel

Gruhlke

2020

Numerical Meth Engineering

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A domain decomposition approach for high-dimensional random partial differential equations exploiting the localization of random parameters is presented. To obtain high efficiency, surrogate models in multielement representations in the parameter space are constructed locally when possible. The method makes use of a stochastic Galerkin finite element tearing interconnecting dual-primal formulation of the underlying problem with localized representations of involved input random fields. Each local parameter space associated to a subdomain is explored by a subdivision into regions where either the parametric surrogate accuracy can be trusted or where instead one has to resort to Monte Carlo. A heuristic adaptive algorithm carries out a problem-dependent hp-refinement in a stochastic multielement sense, anisotropically enlarging the trusted surrogate region as far as possible. This results in an efficient global parameter to solution sampling scheme making use of local parametric smoothness exploration for the surrogate construction. Adequately structured problems for this scheme occur naturally when uncertainties are defined on subdomains, for example, in a multiphysics setting, or when the Karhunen-Loève expansion of a random field can be localized. The efficiency of the proposed hybrid technique is assessed with numerical benchmark problems illustrating the identification of trusted (possibly higher order) surrogate regions and nontrusted sampling regions. K E Y W O R D S domain decomposition, FETI, nonsmooth elliptic partial differential equations, partial differential equations with random coefficients, stochastic finite element method, uncertainty quantification 1 INTRODUCTION In uncertainty quantification (UQ), numerical methods typically are either based on pointwise sampling, which is applicable to quite general problems but rather inefficient, or they rely on (an often analytic) smoothness of the parameter This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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BDDC and FETI-DP under minimalist assumptions

Cited by 24 publications

References 22 publications

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

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

A Unified Analysis of Balancing Domain Decomposition by Constraints for Discontinuous Galerkin Discretizations

A local hybrid surrogate‐based finite element tearing interconnecting dual‐primal method for nonsmooth random partial differential equations

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