A BDDC algorithm with deluxe scaling for $H(curl)$ in two dimensions with irregular subdomains

In FETI-DP (Finite Element Tearing and Interconnecting) and BDDC (Balancing Domain Decomposition by Constraints) domain decomposition methods, the transformation-of-basis approach is used to improve the convergence by combining the local assembly with a change of basis. Suitable basis vectors can be constructed by the recently introduced adaptive coarse space approaches. The resulting FETI-DP and BDDC methods fulfill a condition number bound independent of heterogeneities in the problem. The adaptive method with a transformation of basis presented here builds on a recently introduced adaptive FETI-DP approach for elliptic problems in three dimensions and uses a coarse space constructed from solving small, local eigenvalue problems on closed faces and on a small number of edges. In contrast to our earlier work on adaptive FETI-DP, the coarse space correction is not implemented by using balancing (or deflation), which requires the use of an exact coarse space solver, but by using local transformations. This will make it simpler to extend the method to a large number of subdomains and large supercomputers. The recently established theory of a generalized transformation-of-basis approach yields a condition number estimate for the preconditioned operator that is independent of jumps of the coefficients across and inside subdomains when using the local adaptive constraints. It is shown that all results are also valid for BDDC. Numerical results are presented in three dimensions for FETI-DP and BDDC. We also provide a comparison of different scalings, i.e., deluxe, rho, stiffness, and multiplicity for our adaptive coarse space in 3D.

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“…Last, we consider deluxe-scaling, which was introduced recently; see, e.g, [4,8,10,12,14]. Deluxe-scaling is computationally expensive.…”

mentioning

confidence: 99%

Adaptive FETI-DP and BDDC methods with a generalized transformation of basis for heterogeneous problems

Klawonn¹,

Kühn²,

Rheinbach³

2018

etna

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“…u ; see (3.3). Secondly, we introduce deluxe scaling; see, e.g, [2,6,8,12,13]. Then, the Schur complement of the stiffness matrix on the interface Γ, i.e.,…”

Section: Feti-dp and Bddc With A Transformation Of Basis And Feti-dp mentioning

confidence: 99%

Coarse spaces for FETI-DP and BDDC Methods for heterogeneous problems: connections of deflation and a generalized transformation-of-basis approach

Klawonn¹,

Kühn²,

Rheinbach³

2020

etna

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In FETI-DP (Finite Element Tearing and Interconnecting) and BDDC (Balancing Domain Decomposition by Constraints) domain decomposition methods, the convergence behavior of the iterative scheme can be improved by implementing a coarse space using a transformation of basis and local assembly. This is an alternative to coarse spaces implemented by deflation or balancing. The transformation-of-basis approaches are more robust with respect to inexact solvers than deflation and therefore more suitable for multilevel extensions. In this paper, we show a correspondence of FETI-DP or BDDC methods using a generalized transformation-of-basis approach and of FETI-DP methods using deflation or balancing, where the deflation vectors are obtained from the transformation of basis. These methods then have essentially the same eigenvalues. As opposed to existing theory, this result also applies to general scalings and highly heterogeneous problems. We note that the new methods differ slightly from the classic FETI-DP and BDDC methods using a transformation of basis and that the classic theory has to be replaced. An important application for the theory presented in this paper are FETI-DP and BDDC methods with adaptive coarse spaces, i.e., where deflation vectors are obtained from approximating local eigenvectors. These methods have recently gained considerable interest.

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“…Such scaling has also proven very effective for unstructured grid computation with irregular subdomains given by mesh partitioners [18]. In the deluxe case, the scaling matrices D (i) are block diagonal, with dense blocks corresponding to each class of the local interface.…”

Section: S287mentioning

confidence: 99%

PCBDDC: A Class of Robust Dual-Primal Methods in PETSc

Zampini¹

2016

SIAM J. Sci. Comput.

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Abstract. A class of preconditioners based on balancing domain decomposition by constraints methods is introduced in the Portable, Extensible Toolkit for Scientific Computation (PETSc). The algorithm and the underlying nonoverlapping domain decomposition framework are described with a specific focus on their current implementation in the library. Available user customizations are also presented, together with an experimental interface to the finite element tearing and interconnecting dual-primal methods within PETSc. Large-scale parallel numerical results are provided for the latest version of the code, which is able to tackle symmetric positive definite problems with highly heterogeneous distributions of the coefficients. Current limitations and future extensions of the preconditioner class are also discussed.

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A BDDC algorithm with deluxe scaling for $H(curl)$ in two dimensions with irregular subdomains

Cited by 10 publications

References 36 publications

Adaptive FETI-DP and BDDC methods with a generalized transformation of basis for heterogeneous problems

Adaptive FETI-DP and BDDC methods with a generalized transformation of basis for heterogeneous problems

Coarse spaces for FETI-DP and BDDC Methods for heterogeneous problems: connections of deflation and a generalized transformation-of-basis approach

PCBDDC: A Class of Robust Dual-Primal Methods in PETSc

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