Numerical and computational aspects of some block-preconditioners for saddle point systems

Dorostkar, Ali; Neytcheva, Maya; Lund, Björn

doi:10.1016/j.parco.2015.06.003

Cited by 5 publications

(6 citation statements)

References 41 publications

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“…We note that if A (l) is singular, we can cure this problem by adding a diagonal perturbation of order h 2 , where h is the characteristic discretization parameter. 29,30 Note that diagonal perturbations do not destroy the matrix structure. Define additionally the Boolean matrix…”

Section: Spectral Analysis Of the Cbcs Preconditionermentioning

confidence: 99%

“…Theorem 1. Consider the generalized saddle point linear system (29), where A ∈ R n×n and C ∈ R m×m are symmetric positive definite, B ∈ R m×n has full row rank. Let S e = C + BA −1 B T be the exact Schur complement matrix and S CBC be the inexact cell-by-cell Schur complement matrix defined as in (43).…”

Section: Spectral Analysis Of the Cbcs Preconditionermentioning

confidence: 99%

“…Here, [⋅] denotes the rounding function. The values of n = 2(p + 1) 2 , m = (p + 1) 2 and total number of degrees of freedom (DOF) for the generalized saddle point linear system (29) are listed in Table 2.…”

Section: Implementation Aspectsmentioning

confidence: 99%

“…Recently, in the FEM framework, an approximate Schur complement matrix has been constructed by utilizing the assembly of local element-wise exact Schur complement matrices. [24][25][26][27][28][29][30] That is, we compute the exact Schur complement matrices in each element and then assemble them just like mass or stiffness matrices. The approximate Schur complement matrix obtained in this way remains sparse and has been proved to be a good approximation to the exact one for many problems, for instance the second-order elliptic boundary-value problem, 27 flows in porous media, 28 the Oseen equation, the Cahn-Hilliard equation, 30 and the glacial isostatic adjustment problem.…”

Section: Introductionmentioning

confidence: 99%

“…The approximate Schur complement matrix obtained in this way remains sparse and has been proved to be a good approximation to the exact one for many problems, for instance the second-order elliptic boundary-value problem, 27 flows in porous media, 28 the Oseen equation, the Cahn-Hilliard equation, 30 and the glacial isostatic adjustment problem. 29 In this paper, we extend the idea of the element-by-element Schur complement approximation (also referred to as additive Schur complement approximation) to construct high-quality approximation of S e for the RPIM discretization of the piezoelectric equations. In the RPIM discretization scheme, although only nodes are used to discretize the problem domain, we still need background cells to compute some integrals.…”

Section: Introductionmentioning

confidence: 99%

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Cell‐by‐cell approximate Schur complement technique in preconditioning of meshfree discretized piezoelectric equations

Cao

Neytcheva

2021

Numerical Linear Algebra App

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The radial point interpolation meshfree discretization is a very efficient numerical framework for the analysis of piezoelectricity, in which the fundamental electrostatic equations governing piezoelectric media are solved without mesh generation. Due to the mechanical-electrical coupling property and the piezoelectric constant, the discrete linear system is sparse, of generalized saddle point form and often very ill conditioned. In this work, we propose a technique for constructing a family of cell-by-cell approximate Schur complement matrices, to be used in preconditioning to accelerate the convergence of Krylov subspace iteration methods for such problems. The approximate Schur complement matrices are simply and cheaply constructed in the process of the meshfree discretization and have a sparse structure. It is proved that the so-constructed approximate Schur complement matrices are spectrally equivalent to the exact Schur complement matrix, which leads to very fast convergence when used in preconditioning. In addition, nondimensionalization of the piezoelectric equations is considered to make the computations more stable. The robustness and the efficiency of the proposed preconditioners is illustrated numerically on two test problems, arising from a piezoelectric strip shear deformation problem and a piezoelectric strip bending problem. Numerical results show that the number of iterations to achieve a given tolerance is independent of the number of degrees of freedom as well as of the various problem parameters.

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Section: Spectral Analysis Of the Cbcs Preconditionermentioning

confidence: 99%

Section: Spectral Analysis Of the Cbcs Preconditionermentioning

confidence: 99%

Section: Implementation Aspectsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Cell‐by‐cell approximate Schur complement technique in preconditioning of meshfree discretized piezoelectric equations

Cao

Neytcheva

2021

Numerical Linear Algebra App

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Multidimensional Performance and Scalability Analysis for Diverse Applications Based on System Monitoring Data

Neytcheva

Holmgren

Bull

et al. 2018

Parallel Processing and Applied Mathematics

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Function-based block multigrid strategy for a two-dimensional linear elasticity-type problem

Donatelli

Dorostkar

Mazza

et al. 2017

Computers & Mathematics with Applications

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We consider the solution of block-coupled large-scale linear systems of equations, arising from the finite element approximation of the linear elasticity problem. Due to the large scale of the problems we use properly preconditioned iterative methods, where the preconditioners utilize the underlying block matrix structures, involving inner block solvers and, when suited, broadly established tools such as the algebraic Multigrid method (AMG).For the considered problem, despite of its optimal rate of convergence, AMG, as implemented in some publicly available scientific libraries, imposes unacceptably high demands for computer resources. In this paper we propose and analyze an efficient multilevel preconditioner, based on the Generalized Locally Toeplitz framework, with a specialized transfer operator. We prove and numerically illustrate the optimal convergence rate of the proposed preconditioner, and experimentally report memory and CPU time savings. We also provide comparisons with respect to another aggregation-based algebraic multigrid algorithm

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Numerical and computational aspects of some block-preconditioners for saddle point systems

Cited by 5 publications

References 41 publications

Cell‐by‐cell approximate Schur complement technique in preconditioning of meshfree discretized piezoelectric equations

Cell‐by‐cell approximate Schur complement technique in preconditioning of meshfree discretized piezoelectric equations

Multidimensional Performance and Scalability Analysis for Diverse Applications Based on System Monitoring Data

Function-based block multigrid strategy for a two-dimensional linear elasticity-type problem

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