Adaptive inexact iterative algorithms based on polynomial-degree-robust a posteriori estimates for the Stokes problem
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Abstract
In 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 algorithm, in combination with the Taylor–Hood discretization.
Keywords
Stokes problem Conforming finite element method Adaptive inexact iterative algorithm Outer-inner iteration Uzawa method MinRes A posteriori error estimate Guaranteed bound Efficiency Polynomial-degree-robustness Interplay between error components Adaptive stopping criterionMathematics Subject Classification
65N15 65N22 65N30 65F10 76M10References
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