On a posteriori error bounds for approximations of the generalized Stokes problem generated by the Uzawa algorithm

Anjam, Immanuel; Nokka, Marjaana; Repin, Sergey

doi:10.1515/rnam-2012-0018

Cited by 5 publications

(2 citation statements)

References 7 publications

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“…One of the first contributions balancing algebraic and discretization errors in the Stokes setting is the work of Becker [10]. Later, Silvester and Simoncini [51] derived stopping criteria for the (preconditioned) MinRes algorithm and Anjam et al [3] estimated the total error at each Uzawa step for the exact Uzawa algorithm. Here, one is typically lead to build on a posteriori error estimates obtained for the Stokes problem with an exact algebraic solver; some of the main references here are [55,8,2,39,19,50,57,1,52].…”

Section: Chose Algebraic Solver Starting Approximationsmentioning

confidence: 99%

Adaptive inexact iterative algorithms based on polynomial-degree-robust a posteriori estimates for the Stokes problem

et al. 2017

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International audienceIn this paper, we develop adaptive inexact versions of iterative algorithms applied to finite element discretizations of the linear Stokes problem. We base our developments on an equilibrated stress a posteriori error estimate distinguishing the different error components, namely the discretization error component, the (inner) algebraic solver error component, and possibly the outer algebraic solver error component for algorithms of the Uzawa type. We prove that our estimate gives a guaranteed upper bound on the total error, as well as a polynomial-degree-robust local efficiency, and this on each step of the employed iterative algorithm. Our adaptive algorithms stop the iterations when the corresponding error components do not have a significant influence on the total error. The developed framework covers all standard conforming and conforming stabilized finite element methods on simplicial and rectangular parallelepipeds meshes in two or three space dimensions and an arbitrary algebraic solver. Implementation into the FreeFem++ programming language is invoked and numerical examples showcase the performance of our a posteriori estimates and of the proposed adaptive strategies. As example, we choose here the unpreconditioned and preconditioned Uzawa algorithm and the preconditioned minimum residual (MinRes) algorithm, in combination with the Taylor–Hood discretization

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Section: Chose Algebraic Solver Starting Approximationsmentioning

confidence: 99%

Adaptive inexact iterative algorithms based on polynomial-degree-robust a posteriori estimates for the Stokes problem

et al. 2017

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“…For the generalized Stokes equations, Cahouet and Chabard 31 make a comparison of various iterative solvers based on the preconditioned Uzawa approach. Anjam et al 32 derive a posteriori error bounds for approximations computed by the Uzawa algorithm. The Schur complement of a model problem is considered as a preconditioner for the Uzawa type schemes 33 .…”

Section: Introductionmentioning

confidence: 99%

An Uzawa iterative method for the natural convection problem based on mixed finite element method

Huang

2021

Math Methods in App Sciences

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In this paper, an Uzawa iterative method combined with a mixed finite element method is presented for solving the natural convection model, where a decoupling discrete system is solved and no saddle point system is required to solve at each iterative step. Moreover, the authors establish convergence result of the method and find an interval of relaxation parameter that is crucial for the convergence of the presented method.

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An Uzawa-type algorithm for the coupled Stokes equations

Huang

2020

Appl. Math. Mech.-Engl. Ed.

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On a posteriori error bounds for approximations of the generalized Stokes problem generated by the Uzawa algorithm

Cited by 5 publications

References 7 publications

Adaptive inexact iterative algorithms based on polynomial-degree-robust a posteriori estimates for the Stokes problem

Adaptive inexact iterative algorithms based on polynomial-degree-robust a posteriori estimates for the Stokes problem

An Uzawa iterative method for the natural convection problem based on mixed finite element method

An Uzawa-type algorithm for the coupled Stokes equations

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