Reorthogonalization‐based stiffness preconditioning in FETI algorithms with applications to variational inequalities

Dostál, Zdeněk; Kozubek, Tomáš; Vlach, Oldřich; Brzobohatý, Tomáš

doi:10.1002/nla.1994

Cited by 11 publications

(3 citation statements)

References 32 publications

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“…Both the theoretical results and numerical experiments indicate, rather surprisingly, that unpreconditioned H‐TFETI‐DP with large clusters can be a competitive computational engine for the solution of huge systems of linear equations discretized by sufficiently structured regular grids. Using the reorthogonalization‐based preconditioning, 32 the same performance can be achieved for the problems with variable coefficients provided they are constant on the clusters. The methods of proofs can be used to get some results for more general grids, particularly those obtained by the deformation of a structured grid and for 3D problems.…”

Section: Comments and Conclusionmentioning

confidence: 91%

Bounds on the spectra of Schur complements of large H‐TFETI‐DP clusters for 2D Laplacian

Dostál

Horák

Brzobohatý

et al. 2020

Numerical Linear Algebra App

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Bounds on the spectrum of Schur complements of subdomain stiffness matrices of the discretized Laplacian with respect to interior variables are important in the convergence analysis of finite element tearing and interconnecting (FETI)‐based domain decomposition methods. Here, we are interested in bounds on the regular condition number of Schur complements of “floating” clusters, that is, of matrices comprising the Schur complements of subdomains with prescribed zero Neumann conditions that are joined on the primal level by edge averages. Using some known results, angles of subspaces, and known bounds on the spectrum of Schur complements associated with square domains, we give bounds on the regular condition number of the Schur complement of some “floating” clusters arising from the discretization and decomposition of 2D Laplacian on domains comprising square subdomains. The results show that the condition number of the cluster defined on a fixed domain decomposed into m × m square subdomains joined by edge averages increases proportionally to m. The estimates are compared with numerical values and used in the analysis of H‐FETI‐DP methods. Though the research has been motivated by an effort to extend the scope of scalability of FETI‐based solvers to variational inequalities, the experiments indicate that H‐TFETI‐DP with large clusters can be useful for the solution of huge linear elliptic problems discretized by sufficiently regular grids.

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Section: Comments and Conclusionmentioning

confidence: 91%

Bounds on the spectra of Schur complements of large H‐TFETI‐DP clusters for 2D Laplacian

Dostál

Horák

Brzobohatý

et al. 2020

Numerical Linear Algebra App

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“…

Šístek et al[2] deal with the solution of saddle point systems arising from the mixed-hybrid finite element discretization of flow in porous media including highly permeable fractures. The fractures are modeled by 1D or 2D elements inside three-dimensional domains.

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confidence: 99%

“…The collection of five papers [2][3][4][5][6] published in this issue of Numerical Linear Algebra with Applications originates from the talks delivered at the conference Preconditioning of Iterative MethodsTheory and Applications (PIM 2013), organized in honor of Professor Ivo Marek on the occasion of his 80th birthday and represents a tribute to his work and influence in the numerical linear algebra community, compare with [1].…”

mentioning

confidence: 99%

Preconditioning of iterative methods ‐ theory and applications

Axelsson

Blaheta

Neytcheva

et al. 2015

Numerical Linear Algebra App

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The collection of five papers [2][3][4][5][6] published in this issue of Numerical Linear Algebra with Applications originates from the talks delivered at the conference Preconditioning of Iterative MethodsTheory and Applications (PIM 2013), organized in honor of Professor Ivo Marek on the occasion of his 80th birthday and represents a tribute to his work and influence in the numerical linear algebra community, compare with [1].We provide a short summary of the presented papers in order of their submission. Šístek et al.[2] deal with the solution of saddle point systems arising from the mixed-hybrid finite element discretization of flow in porous media including highly permeable fractures. The fractures are modeled by 1D or 2D elements inside three-dimensional domains. The paper deals with an application of the balancing domain decomposition by constraints method to the described problems and provides both theoretical results and numerical experiments, which illustrate the efficiency and the scalability of the balancing domain decomposition by constraints method.Axelsson et al.[3] consider a particular type of a high-order strongly stable time integration method that is applicable to evolution and differential-algebraic problems. To solve the block matrix systems arising within the corresponding time stepping procedure, an efficient preconditioning method is presented and analyzed. The method is applied to a consolidation problem arising in poroelasticity. Kužel and Vaněk [4] propose a special type of nonlinear multigrid scheme in which the prolongator (P ) is updated at each iteration so that the current approximation lies in range of P . The update of the prolongator is carried out in a simple way by adding the current approximation as a new column in the prolongation matrix. An advantage of this scheme constitutes improved convergence with easy implementation. The method is applicable for the solution of both linear and nonlinear problems. In particular, the paper describes application to generalized eigenvalue problems.Kraus et al.[5] present a non-variational auxiliary space multigrid algorithm for general symmetric positive definite problems. The method is based on exact two-by-two block factorization of local (finite element) matrices that correspond to a sequence of coverings of the domain by overlapping or non-overlapping subdomains. The coarse grid matrix is defined as an additive-type approximation of the Schur complement. Its sparsity is controlled by the size and overlap of the subdomains. The coarse grid construction is used in the framework of the Algebraic Multilevel Iterative (AMLI) method, and the efficiency and the robustness of the method when solving multiscale problems with oscillating coefficients are shown theoretically as well as by numerical experiments.Dostál et al.

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Hybrid TBETI domain decomposition for huge 2D scalar variational inequalities

Dostál,

Sadowská,

Horák

et al. 2024

Numerical Meth Engineering

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The unpreconditioned H‐TFETI‐DP (hybrid total finite element tearing and interconnecting dual‐primal) domain decomposition method introduced by Klawonn and Rheinbach turned out to be an effective solver for variational inequalities discretized by huge structured grids. The basic idea is to decompose the domain into non‐overlapping subdomains, interconnect some adjacent subdomains into clusters on a primal level, and enforce the continuity of the solution across both the subdomain and cluster interfaces by Lagrange multipliers. After eliminating the primal variables, we get a reasonably conditioned quadratic programming (QP) problem with bound and equality constraints. Here, we first reduce the continuous problem to the subdomains' boundaries, then discretize it using the boundary element method, and finally interconnect the subdomains by the averages of adjacent edges. The resulting QP problem in multipliers with a small coarse grid is solved by specialized QP algorithms with optimal complexity. The method can be considered as a three‐level multigrid with the coarse grids split between primal and dual variables. Numerical experiments illustrate the efficiency of the presented H‐TBETI‐DP (hybrid total boundary element tearing and interconnecting dual‐primal) method and nice spectral properties of the discretized Steklov–Poincaré operators as compared with their finite element counterparts.

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Reorthogonalization‐based stiffness preconditioning in FETI algorithms with applications to variational inequalities

Cited by 11 publications

References 32 publications

Bounds on the spectra of Schur complements of large H‐TFETI‐DP clusters for 2D Laplacian

Bounds on the spectra of Schur complements of large H‐TFETI‐DP clusters for 2D Laplacian

Preconditioning of iterative methods ‐ theory and applications

Hybrid TBETI domain decomposition for huge 2D scalar variational inequalities

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