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

Abstract

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

Stable broken $H^1$ and $H(\mathrm {div})$ polynomial extensions for polynomial-degree-robust potential and flux reconstruction in three space dimensions

“…Another extension of the present work would be to cover the H(curl) case, hinging on the single tetrahedron results of Demkowicz et al [12,Theorem 7.2]. We also mention that the application of the present results to the construction of p-robust a posteriori error estimates for problems with arbitrarily jumping coefficients is detailed in [9], to eigenvalue problems in [5,6], to the Stokes problem in [8], to linear elasticity in [17], and to the heat equation in [19,20].…”

Krylov improvements of the Uzawa method for Stokes type operator matrices

“…Uzawa iteration A popular way to solve the Stokes problem is the Uzawa iteration, which is still in the focus of recent interest, see, e.g., [16,20,24]. On the continuous level and in strong form, it reads as…”

Equilibrated Stress Tensor Reconstruction and A Posteriori Error Estimation for Nonlinear Elasticity

“…Therefore, its implementation is independent and directly applicable to these laws, which makes the method convenient for FEM softwares in solid mechanics, which often provide a large choice of behavior laws. In addition, equilibrated error estimates were proven to be polynomial-degree robust for several linear problems in 2D, as the Poisson problem in [7,18], linear elasticity in [14] and the related Stokes problem in [10] and recently in 3D in [19]. This paper is organized as follows.…”