On a hierarchical error estimator combined with a stabilized method for the Navier–Stokes equations

Abstract

This work combines two complementary strategies for solving the steady incompressible Navier-Stokes model with a zeroth-order term, namely, a stabilized finite element method and a mesh-refinement approach based on an error estimator. First, equal order interpolation spaces are adopted to approximate both the velocity and the pressure while stability is recovered within the stabilization approach. Also designed to handle advection dominated flows under zeroth-order term influence, the stabilized method incorporates a new parameter with a threefold asymptotic behavior. Mesh adaptivity driven by a new hierarchical error estimator and built on the stabilized method is the second ingredient. The estimator construction process circumvents the saturation assumption by using an enhancing space strategy which is shown to be equivalent to the error. Several numerical tests validate the methodology.

A Posteriori Estimates of Taylor-Hood Element for Stokes Problem Using Auxiliary Subspace Techniques

“…The error analysis of upper and lower bounds avoids the use of saturation assumption. Although the construction of auxiliary space needs a transformation operator in the reference element, it provides a novel idea for removing saturation assumption in reliability analysis [2,3,4,5]. Hakula et al constructed the auxiliary space directly on each element for the second order elliptic problem and elliptic eigenvalue problem and proved that the error is bounded by the error estimator up to oscillation terms without the saturation assumption [17,15].…”

An a Posteriori Error Estimate for Mixed Finite Element Approximations of the Navier-Stokes Equations

“…In particular, for the Stokes problem, M. Ainsworth and J. Oden [2], D. Kay and D. Silvester [23], C. Cartensen and S. A. Fuken [9] and R. Verfurth [31] introduced several error estimators and provided that they are equivalent to the energy norm of the errors. Other works for the stationary Navier-Stokes problem have been introduced in [27], [32], [22], [34], [1], [26], [3].…”