A posteriori error estimators for a two-level finite element method for the Navier-Stokes equations

Abstract

Two-and multilevel truncated Newton finite element discretizations are presently a very promising approach for approximating the (nonlinear) Navier-Stokes equations describing the equilibrium flow of a viscous, incompressible fluid. Their combination with mesh adaptivity is considered in this article. Specifically, locally calculable c1 posteriori error estimators are derived, with full mathematical support, for the basic two-level discretization of the Navier-Stokes equations. 0 1996

Residual a Posteriori Error Estimate of a New Two-Level Method for Steady Navier-Stokes Equations

“…Thus, the extra terms have to be computed in practice to guarantee an asymptotically correct a posteriori error estimates. To our knowledge, only in [1] and [7], the authors combine two-level method with adaptive grid refinement, but in this article, we introduce a new two-level method being different from [1] and [7] and study its a posteriori error estimate serving to control an adaptive grid refinement.…”

“…Theorem 3.1 [1,7] If h is sufficient small, then (u h , p h ) generated by (4) satisfies a posteriori error estimate as follows:…”

“…Theorem 3.2 [1,7] With the assumption (8) on the stability of the linearized dual problem (7), under the condition of Theorem 3.1, there is a posteriori error estimate as follows:…”

An efficient two-step algorithm for the incompressible flow problem

“…Hence, it can be seen as a two-step predictor-corrector procedure and can save a large amount of computation compared to the one-level methods. Some details of the two-level method can be found in the works of Xu [31,32], Ammi and Marion [1], Layton et al [20][21][22], Zhang and He [33], Ervin et al [6], He et al [10,11,13] and Li [24].…”

Two-level stabilized nonconforming finite element method for the Stokes equations

“…Some details on the two-level approach can be found in the papers of He [4], Xu [12], [13], Layton [9], Layton and Lenferink [7], Ervin et al [2], Layton and Tobiska [8].…”